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Journal articles on the topic 'Bounded Symmetric domain'

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Published: 6 September 2023

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1

ROOS, Guy. "Bergman–Hartogs domains and their automorphisms." Tambov University Reports. Series: Natural and Technical Sciences, no. 127 (2019): 316–23. http://dx.doi.org/10.20310/2686-9667-2019-24-127-316-323.

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For Cartan–Hartogs domains and also for Bergman–Hartogs domains, the determination of their automorphism groups is given for the cases when the base is any bounded symmetric domain and a general bounded homogeneous domain respectively.

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2

BERSHTEIN, OLGA. "REGULAR FUNCTIONS ON THE SHILOV BOUNDARY." Journal of Algebra and Its Applications 04, no. 06 (December 2005): 613–29. http://dx.doi.org/10.1142/s0219498805001447.

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In this paper a *-algebra of regular functions on the Shilov boundary S(𝔻) of bounded symmetric domain 𝔻 is constructed. The algebras of regular functions on S(𝔻) are described in terms of generators and relations for two particular series of bounded symmetric domains. Also, the degenerate principal series of quantum Harish–Chandra modules related to S(𝔻) = Un is investigated.

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3

Ramachandran, C., R. Ambrose Prabhu, and Srikandan Sivasubramanian. "Starlike and convex functions with respect to symmetric conjugate points involving conical domain." Mathematica Slovaca 68, no. 1 (February 23, 2018): 89–102. http://dx.doi.org/10.1515/ms-2017-0083.

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AbstractEnough attentions to domains related to conical sections has not been done so far although it deserves more. Making use of the conical domain the authors have defined a new class of starlike and Convex Functions with respect to symmetric points involving the conical domain. Growth and distortion estimates are studied with convolution using domains bounded by conic regions. Certain coefficient estimates are obtained for domains bounded by conical region. Finally interesting application of the results are also highlighted for the function Ωk,βdefined by Noor.

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4

Di Scala, Antonio J., Andrea Loi, and Guy Roos. "The Bisymplectomorphism Group of a Bounded Symmetric Domain." Transformation Groups 13, no. 2 (June 2008): 283–304. http://dx.doi.org/10.1007/s00031-008-9015-z.

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5

Mackey and Mellon. "The Bergmann-Shilov boundary of a Bounded Symmetric Domain." Mathematical Proceedings of the Royal Irish Academy 121A, no. 2 (2021): 33. http://dx.doi.org/10.3318/pria.2021.121.03.

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6

Mackey, M., and P. Mellon. "The Bergmann-Shilov boundary of a Bounded Symmetric Domain." Mathematical Proceedings of the Royal Irish Academy 121, no. 2 (2021): 33–49. http://dx.doi.org/10.1353/mpr.2021.0002.

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7

Choi, Ki-Seong. "NOTES ON CARLESON TYPE MEASURES ON BOUNDED SYMMETRIC DOMAIN." Communications of the Korean Mathematical Society 22, no. 1 (January 31, 2007): 65–74. http://dx.doi.org/10.4134/ckms.2007.22.1.065.

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8

PAN, LISHUANG, AN WANG, and LIYOU ZHANG. "ON THE KÄHLER–EINSTEIN METRIC OF BERGMAN–HARTOGS DOMAINS." Nagoya Mathematical Journal 221, no. 1 (March 2016): 184–206. http://dx.doi.org/10.1017/nmj.2016.4.

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We study the complete Kähler–Einstein metric of certain Hartogs domains ${\rm\Omega}_{s}$ over bounded homogeneous domains in $\mathbb{C}^{n}$. The generating function of the Kähler–Einstein metric satisfies a complex Monge–Ampère equation with Dirichlet boundary condition. We reduce the Monge–Ampère equation to an ordinary differential equation and solve it explicitly when we take the parameter $s$ for some critical value. This generalizes previous results when the base is either the Euclidean unit ball or a bounded symmetric domain.

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9

Neidhardt, H., and V. A. Zagrebnov. "Does Each Symmetric Operator Have a Stability Domain?" Reviews in Mathematical Physics 10, no. 06 (August 1998): 829–50. http://dx.doi.org/10.1142/s0129055x98000276.

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We show that any symmetric operator H has a dense maximal b-stability domain [Formula: see text] (i.e. [Formula: see text], b∈R1) if and only if H is unbounded from above. This abstract result allows an application to singular perturbed Schrödinger operators which are not semi-bounded from below, i.e., to the so-called "fall to the center problem". It turns out that in this case the regularization problem is always ill-posed which implies that there is no unique "right Hamiltonian" for corresponding perturbed system. We give an example of singular perturbed Schrödinger operator for which stability domains are described explicitly.

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10

Wu, Hio Tong, Ieng Tak Leong, and Tao Qian. "Adaptive rational approximation in Bergman space on bounded symmetric domain." Journal of Mathematical Analysis and Applications 506, no. 1 (February 2022): 125591. http://dx.doi.org/10.1016/j.jmaa.2021.125591.

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11

Chu, Cho-Ho, Hidetaka Hamada, Tatsuhiro Honda, and Gabriela Kohr. "Bloch Space of a Bounded Symmetric Domain and Composition Operators." Complex Analysis and Operator Theory 13, no. 2 (August 21, 2018): 479–92. http://dx.doi.org/10.1007/s11785-018-0835-0.

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12

Zhu, Fuliu. "The inverse Abel transform for an exceptional bounded symmetric domain." Science in China Series A: Mathematics 40, no. 7 (July 1997): 687–96. http://dx.doi.org/10.1007/bf02878691.

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13

Chu, Cho-Ho. "Siegel domains over Finsler symmetric cones." Journal für die reine und angewandte Mathematik (Crelles Journal) 2021, no. 778 (May 29, 2021): 145–69. http://dx.doi.org/10.1515/crelle-2021-0027.

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Abstract Let Ω be a proper open cone in a real Banach space V. We show that the tube domain V ⊕ i ⁢ Ω {V\oplus i\Omega} over Ω is biholomorphic to a bounded symmetric domain if and only if Ω is a normal linearly homogeneous Finsler symmetric cone, which is equivalent to the condition that V is a unital JB-algebra in an equivalent norm and Ω is the interior of { v 2 : v ∈ V } {\{v^{2}:v\in V\}} .

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14

Wolniewicz, Tomasz M. "Independent inner functions in the classical domains." Glasgow Mathematical Journal 29, no. 2 (July 1987): 229–36. http://dx.doi.org/10.1017/s001708950000687x.

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Let Bn denote the unit ball and Un the unit polydisc in Cn. In this paper we consider questions concerned with inner functions and embeddings of Hardy spaces over bounded symmetric domains. The main result (Theorem 2) states that for a classical symmetric domain D of type I and rank(D) = s, there exists an isometric embedding of Hl(Us) onto a complemented subspace of Hl(D). This should be compared with results of Wojtaszczyk [13] and Bourgain [3, 4] which state that H1(Bn) is isomorphic to Hl(U) while for n>m, Hl(Un) cannot be isomorphically embedded onto a complemented subspace of H1(Um). Since balls are the only bounded symmetric domains of rank 1 and they are of type I, Theorem 2 shows that if rank(D1) = 1, rank(D2) > 1 then H1(D1) is not isomorphic to H1(D2). It is natural to expect this to hold always when rank(D1 ≠ rank(D2) but unfortunately we were not able to prove this.

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15

Hess, Peter, and P. Poláčik. "Symmetry and convergence properties for non-negative solutions of nonautonomous reaction–diffusion problems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 124, no. 3 (1994): 573–87. http://dx.doi.org/10.1017/s030821050002878x.

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Nonautonomous parabolic equations of the form ut − Δu = f(u, t) on a symmetric domain are considered. Using the moving-hyperplane method, it is proved that any bounded nonnegative solution symmetrises as t → ∞. This is then used to show that for nonlinearities periodic in t, any non-negative bounded solution approaches a periodic solution.

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16

XU, Qin. "Completeness of Normal Modes for Symmetric Perturbations in Vertically Bounded Domain." Journal of the Meteorological Society of Japan 89, no. 4 (2011): 389–94. http://dx.doi.org/10.2151/jmsj.2011-407.

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17

Honda, Tatsuhiro. "Operators of the alfa-Bloch space on the open unit ball of a JB*-triple." Studia Universitatis Babes-Bolyai Matematica 67, no. 2 (June 8, 2022): 317–28. http://dx.doi.org/10.24193/subbmath.2022.2.08.

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"Let $\B_X$ be a bounded symmetric domain realized as the open unit ball of a JB*-triple $X$ which may be infinite dimensional. In this paper, we characterize the bounded weighted composition operators from the Hardy space $H^{\infty}(\mathbb{B}_X)$ into the $\alpha $-Bloch space $\mathcal{B}^\alpha (\B_X)$ on $\mathbb{B}_X$. Later, we show the multiplication operator from $H^{\infty}(\mathbb{B}_X)$ into $\mathcal{B}^\alpha (\B_X)$ is bounded. Also, we give the operator norm of the bounded multiplication operator."

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18

Mellon, P. "Dynamics of Biholomorphic Self-Maps on Bounded Symmetric Domains." MATHEMATICA SCANDINAVICA 117, no. 2 (December 14, 2015): 203. http://dx.doi.org/10.7146/math.scand.a-22867.

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Let $g$ be a fixed-point free biholomorphic self-map of a bounded symmetric domain $B$. It is known that the sequence of iterates $(g^n)$ may not always converge locally uniformly on $B$ even, for example, if $B$ is an infinite dimensional Hilbert ball. However, $g=g_a\circ T$, for a linear isometry $T$, $a=g(0)$ and a transvection $g_a$, and we show that it is possible to determine the dynamics of $g_a$. We prove that the sequence of iterates $(g_a^n)$ converges locally uniformly on $B$ if, and only if, $a$ is regular, in which case, the limit is a holomorphic map of $B$ onto a boundary component (surprisingly though, generally not the boundary component of $\frac{a}{\|a\|}$). We prove $(g_a^n)$ converges to a constant for all non-zero $a$ if, and only if, $B$ is a complex Hilbert ball. The results are new even in finite dimensions where every element is regular.

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19

Kajikiya, Ryuji. "Symmetric and Asymmetric Solutions of p-Laplace Elliptic Equations in Hollow Domains." Advanced Nonlinear Studies 18, no. 2 (April 1, 2018): 303–21. http://dx.doi.org/10.1515/ans-2017-6023.

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AbstractIn the present paper, we study thep-Laplace equation in a hollow symmetric bounded domain. LetHandGbe closed subgroups of the orthogonal group such that{H\varsubsetneq G}. Then we prove the existence of a positive solution which isH-invariant andG-non-invariant. Furthermore, we give several examples ofH,Gand Ω, and find symmetric and asymmetric solutions.

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20

WILKINS, DAVID R. "HOMOGENEOUS VECTOR BUNDLES AND COWEN-DOUGLAS OPERATORS." International Journal of Mathematics 04, no. 03 (June 1993): 503–20. http://dx.doi.org/10.1142/s0129167x93000261.

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In this paper we obtain an algebraic classification of all homogeneous Hermitian holomorphic vector bundles of arbitrary rank over a bounded symmetric domain. This classification result is used in order to classify, up to unitary equivalence, all irreducible homogeneous bounded linear operators on a separable infinite-dimensional Hilbert space that belong to the Cowen-Douglas class B2 (∆), where ∆ is the open unit disk.

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21

Fattahi, Fariba, and Mohsen Alimohammady. "Infinitely many solutions for a class of hemivariational inequalities involving p(x)-Laplacian." Analele Universitatii "Ovidius" Constanta - Seria Matematica 25, no. 2 (July 26, 2017): 65–83. http://dx.doi.org/10.1515/auom-2017-0021.

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AbstractIn this paper hemivariational inequality with nonhomogeneous Neumann boundary condition is investigated. The existence of infinitely many small solutions involving a class of p(x) - Laplacian equation in a smooth bounded domain is established. Our main tool is based on a version of the symmetric mountain pass lemma due to Kajikiya and the principle of symmetric criticality for a locally Lipschitz functional.

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22

Clapp, Mónica, Manuel Del Pino, and Monica Musso. "Multiple solutions for a non-homogeneous elliptic equation at the critical exponent." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 134, no. 1 (February 2004): 69–87. http://dx.doi.org/10.1017/s0308210500003085.

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We consider the equation−Δu = |u|4/(N−2)u + εf(x) under zero Dirichlet boundary conditions in a bounded domain Ω in RN exhibiting certain symmetries, with f ≥ 0, f ≠ 0. In particular, we find that the number of sign-changing solutions goes to infinity for radially symmetric f, as ε → 0 if Ω is a ball. The same is true for the number of negative solutions if Ω is an annulus and the support of f is compact in Ω.

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23

Friedman, Yaakov, and Tzvi Scarr. "Symmetry and Special Relativity." Symmetry 11, no. 10 (October 3, 2019): 1235. http://dx.doi.org/10.3390/sym11101235.

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We explore the role of symmetry in the theory of Special Relativity. Using the symmetry of the principle of relativity and eliminating the Galilean transformations, we obtain a universally preserved speed and an invariant metric, without assuming the constancy of the speed of light. We also obtain the spacetime transformations between inertial frames depending on this speed. From experimental evidence, this universally preserved speed is c, the speed of light, and the transformations are the usual Lorentz transformations. The ball of relativistically admissible velocities is a bounded symmetric domain with respect to the group of affine automorphisms. The generators of velocity addition lead to a relativistic dynamics equation. To obtain explicit solutions for the important case of the motion of a charged particle in constant, uniform, and perpendicular electric and magnetic fields, one can take advantage of an additional symmetry—the symmetric velocities. The corresponding bounded domain is symmetric with respect to the conformal maps. This leads to explicit analytic solutions for the motion of the charged particle.

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24

Matsumoto, Keiji. "Theta functions on the classical bounded symmetric domain of type $I_{2,2}$." Proceedings of the Japan Academy, Series A, Mathematical Sciences 67, no. 1 (1991): 1–5. http://dx.doi.org/10.3792/pjaa.67.1.

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25

Clerc, Jean-Louis, and Karl-Hermann Neeb. "Orbits of triples in the Shilov boundary of a bounded symmetric domain." Transformation Groups 11, no. 3 (September 2006): 387–426. http://dx.doi.org/10.1007/s00031-005-1117-2.

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26

Meschiari, Mauro. "Proper holomorphic maps on an irreducible bounded symmetric domain of classical type." Rendiconti del Circolo Matematico di Palermo 37, no. 1 (February 1988): 18–34. http://dx.doi.org/10.1007/bf02844266.

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27

Nomura, Takaaki. "Algebraically independent generators of invariant differential operators on a bounded symmetric domain." Journal of Mathematics of Kyoto University 31, no. 1 (1991): 265–79. http://dx.doi.org/10.1215/kjm/1250519904.

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28

Ren, Shuzhan. "Normal modes in the symmetric stability problem in a vertically bounded domain." Geophysical & Astrophysical Fluid Dynamics 102, no. 4 (August 2008): 333–48. http://dx.doi.org/10.1080/03091920701841606.

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29

Faraut, Jacques, and Masato Wakayama. "Hermitian symmetric spaces of tube type and multivariate Meixner-Pollaczek polynomials." MATHEMATICA SCANDINAVICA 120, no. 1 (February 23, 2017): 87. http://dx.doi.org/10.7146/math.scand.a-25506.

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Harmonic analysis on Hermitian symmetric spaces of tube type is a natural framework for introducing multivariate Meixner-Pollaczek polynomials. Their main properties are established in this setting: orthogonality, generating and determinantal formulae, difference equations. For proving these properties we use the composition of the following transformations: Cayley transform, Laplace transform, and spherical Fourier transform associated to Hermitian symmetric spaces of tube type. In particular the difference equation for the multivariate Meixner-Pollaczek polynomials is obtained from an Euler type equation on a bounded symmetric domain.

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30

Boman, Jan. "A hypersurface containing the support of a Radon transform must be an ellipsoid. II: The general case." Journal of Inverse and Ill-posed Problems 29, no. 3 (February 2, 2021): 351–67. http://dx.doi.org/10.1515/jiip-2020-0139.

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Abstract If the Radon transform of a compactly supported distribution f ≠ 0 {f\neq 0} in ℝ n {\mathbb{R}^{n}} is supported on the set of tangent planes to the boundary ∂ ⁡ D {\partial D} of a bounded convex domain D, then ∂ ⁡ D {\partial D} must be an ellipsoid. The special case of this result when the domain D is symmetric was treated in [J. Boman, A hypersurface containing the support of a Radon transform must be an ellipsoid. I: The symmetric case, J. Geom. Anal. 2020, 10.1007/s12220-020-00372-8]. Here we treat the general case.

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31

Iacopetti, Alessandro, and Giusi Vaira. "Sign-changing tower of bubbles for the Brezis–Nirenberg problem." Communications in Contemporary Mathematics 18, no. 01 (January 29, 2016): 1550036. http://dx.doi.org/10.1142/s0219199715500364.

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In this paper, we prove that the Brezis–Nirenberg problem: [Formula: see text] where [Formula: see text] is a symmetric bounded smooth domain in [Formula: see text], [Formula: see text] and [Formula: see text], has a solution with the shape of a tower of two bubbles with alternate signs, centered at the center of symmetry of the domain, for all [Formula: see text] sufficiently small.

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32

Xu, Qin. "Modal and Nonmodal Growths of Symmetric Perturbations in Unbounded Domain." Journal of the Atmospheric Sciences 67, no. 6 (June 1, 2010): 1996–2017. http://dx.doi.org/10.1175/2010jas3360.1.

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Abstract Modal and nonmodal growths of nonhydrostatic symmetric perturbations in an unbounded domain are examined in comparison with their hydrostatic counterparts. It is shown that the modal growth rate is a function of a single internal parameter s, the slope of the cross-band wave pattern. The maximum nonmodal growth of total perturbation energy norm is produced, also as a function of s, by an optimal combination of one geostrophic neutral mode and two paired nongeostrophic growing and decaying (or propagating) modes in the unstable (or stable) region. The hydrostatic approximation inflates the maximum modal growth rate significantly (or boundlessly) as the basic-state Richardson number Ri is small (or → 0) and inflates the maximum nonmodal growth rate significantly (or boundlessly) as |s| is large (or → ∞). Inside the unstable region, the maximum nonmodal growth scaled by the modal growth is a bounded increasing function of growth time τ but reduces to 1 at (Ri, s) = (¼, −2) where the three modes become orthogonal to each other. Outside the unstable region, the maximum nonmodal growth is a periodic function of τ and the maximum growth time τm is bounded between ¼ and ½ of the period of the paired propagating modes. The scaled maximum nonmodal growth reaches the global maximum at s = −Ri−1 ± Ri−1(1 − Ri)1/2 (the marginal-stability boundary) for any τ if Ri ≤ 1, or at s = −1 ± (1 − Ri−1)1/2 for τ = τm if Ri > 1. When the neutral mode is filtered, the nonmodal growth becomes nongeostrophic and smaller than its counterpart growth constructed by the three modes but still significantly larger than the modal growth in general. The scaled maximum nongeostrophic nonmodal growth reaches the global maximum at s = −Ri−1 ± Ri−1(1 − Ri)1/2 for any τ if Ri ≤ 1, or at s = −Ri−1/2 for τ = τm if Ri > 1. Normalized inner products between the modes are introduced to measure their nonorthogonality and interpret their constructed nonmodal growths physically.

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33

Ohnawa, Masashi, and Masahiro Suzuki. "Time-periodic solutions of symmetric hyperbolic systems." Journal of Hyperbolic Differential Equations 17, no. 04 (December 2020): 707–26. http://dx.doi.org/10.1142/s0219891620500216.

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We prove the unique existence of time-periodic solutions to general hyperbolic equations with periodic external forces autonomous or nonautonomous over a domain bounded by two parallel planes, provided that all the characteristics with respect to the direction normal to the planes have the same sign. It is also shown that global-in-time solutions to initial-boundary value problems coincide with the solutions to corresponding time-periodic problems after a finite time. We devote one section to the reformulation of several realistic problems and see our results have wide applicability.

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34

Clerc, Jean-Louis. "An invariant for triples in the Shilov boundary of a bounded symmetric domain." Communications in Analysis and Geometry 15, no. 1 (2007): 147–74. http://dx.doi.org/10.4310/cag.2007.v15.n1.a5.

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35

Mahmood, Shahid, Hari Srivastava, and Sarfraz Malik. "Some Subclasses of Uniformly Univalent Functions with Respect to Symmetric Points." Symmetry 11, no. 2 (February 22, 2019): 287. http://dx.doi.org/10.3390/sym11020287.

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This article presents the study of certain analytic functions defined by bounded radius rotations associated with conic domain. Many geometric properties like coefficient estimate, radii problems, arc length, integral representation, inclusion results and growth rate of coefficients of Taylor’s series representation are investigated. By varying the parameters in results, several well-known results in literature are obtained as special cases.

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36

Dong, Yan. "Study of Weak Solutions for Degenerate Parabolic Inequalities with Nonlocal Nonlinearities." Symmetry 14, no. 8 (August 13, 2022): 1683. http://dx.doi.org/10.3390/sym14081683.

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This paper studies a class of variational inequalities with degenerate parabolic operators and symmetric structure, which is an extension of the parabolic equation in a bounded domain. By solving a series of penalty problems, the existence and uniqueness of the solutions in the weak sense are proved by the energy method and a limit process.

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37

Žubrinić, Darko. "Solvability of quasilinear elliptic equations with strong dependence on the gradient." Abstract and Applied Analysis 5, no. 3 (2000): 159–73. http://dx.doi.org/10.1155/s1085337500000324.

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We study the problem of existence of positive, spherically symmetric strong solutions of quasilinear elliptic equations involvingp-Laplacian in the ball. We allow simultaneous strong dependence of the right-hand side on both the unknown function and its gradient. The elliptic problem is studied by relating it to the corresponding singular ordinary integro-differential equation. Solvability range is obtained in the form of simple inequalities involving the coefficients describing the problem. We also study a posteriori regularity of solutions. An existence result is formulated for elliptic equations on arbitrary bounded domains in dependence of outer radius of domain.

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38

Khan, Muhammmad Ghaffar, Wali Khan Mashwani, Lei Shi, Serkan Araci, Bakhtiar Ahmad, and Bilal Khan. "Hankel inequalities for bounded turning functions in the domain of cosine Hyperbolic function." AIMS Mathematics 8, no. 9 (2023): 21993–2008. http://dx.doi.org/10.3934/math.20231121.

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<abstract><p>In the present article, we define and investigate a new subfamily of holomorphic functions connected with the cosine hyperbolic function with bounded turning. Further some interesting results like sharp coefficients bounds, sharp Fekete-Szegö estimate, sharp $ 2^{nd} $ Hankel determinant and non-sharp $ 3^{rd} $ order Hankel determinant. Moreover, the same estimates have been investigated for 2-fold, 3-fold symmetric functions, the first four initial sharp bounds of logarithmic coefficient and sharp second Hankel determinant of logarithmic coefficients fort his defined function family.</p></abstract>

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39

Clark, Marcondes Rodrigues. "Existence of solutions for a nonlinear hyperbolic-parabolic equation in a non-cylinder domain." International Journal of Mathematics and Mathematical Sciences 19, no. 1 (1996): 151–60. http://dx.doi.org/10.1155/s0161171296000221.

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In this paper, we study the existence of global weak solutions for the equationk2(x)u″+k1(x)u′+A(t)u+|u|ρu=f (I)in the non-cylinder domainQinRn+1;k1andk2are bounded real functions,A(t)is the symmetric operatorA(t)=−∑i,j=1n∂∂xj(aij(x,t)∂∂xi) whereaijandfare real functions given inQ. For the proof of existence of global weak solutions we use the Faedo-Galerkin method, compactness arguments and penalization.

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40

Endres, Erik, Helge Kristian Jenssen, and Mark Williams. "Symmetric Euler and Navier–Stokes shocks in stationary barotropic flow on a bounded domain." Journal of Differential Equations 245, no. 10 (November 2008): 3025–67. http://dx.doi.org/10.1016/j.jde.2008.03.013.

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41

Bañuelos, Rodrigo, and Dante DeBlassie. "On the First Eigenfunction of the Symmetric Stable Process in a Bounded Lipschitz Domain." Potential Analysis 42, no. 2 (October 18, 2014): 573–83. http://dx.doi.org/10.1007/s11118-014-9445-2.

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42

Yang, Sibei, Dachun Yang, and Wen Yuan. "Global gradient estimates for Dirichlet problems of elliptic operators with a BMO antisymmetric part." Advances in Nonlinear Analysis 11, no. 1 (January 1, 2022): 1496–530. http://dx.doi.org/10.1515/anona-2022-0247.

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Abstract Let n ≥ 2 n\ge 2 and Ω ⊂ R n \Omega \subset {{\mathbb{R}}}^{n} be a bounded nontangentially accessible domain. In this article, the authors investigate (weighted) global gradient estimates for Dirichlet boundary value problems of second-order elliptic equations of divergence form with an elliptic symmetric part and a BMO antisymmetric part in Ω \Omega . More precisely, for any given p ∈ ( 2 , ∞ ) p\in \left(2,\infty ) , the authors prove that a weak reverse Hölder inequality with exponent p p implies the global W 1 , p {W}^{1,p} estimate and the global weighted W 1 , q {W}^{1,q} estimate, with q ∈ [ 2 , p ] q\in \left[2,p] and some Muckenhoupt weights, of solutions to Dirichlet boundary value problems. As applications, the authors establish some global gradient estimates for solutions to Dirichlet boundary value problems of second-order elliptic equations of divergence form with small BMO {\rm{BMO}} symmetric part and small BMO {\rm{BMO}} antisymmetric part, respectively, on bounded Lipschitz domains, quasi-convex domains, Reifenberg flat domains, C 1 {C}^{1} domains, or (semi-)convex domains, in weighted Lebesgue spaces. Furthermore, as further applications, the authors obtain the global gradient estimate, respectively, in (weighted) Lorentz spaces, (Lorentz–)Morrey spaces, (Musielak–)Orlicz spaces, and variable Lebesgue spaces. Even on global gradient estimates in Lebesgue spaces, the results obtained in this article improve the known results via weakening the assumption on the coefficient matrix.

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43

Helffer, B., T. Hoffmann-Ostenhof, F. Jauberteau, and C. Léna. "On the multiplicity of the second eigenvalue of the Laplacian in non simply connected domains – with some numerics –." Asymptotic Analysis 121, no. 1 (December 2, 2020): 35–57. http://dx.doi.org/10.3233/asy-191594.

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We revisit an interesting example proposed by Maria Hoffmann-Ostenhof, the second author and Nikolai Nadirashvili of a bounded domain in R 2 for which the second eigenvalue of the Dirichlet Laplacian has multiplicity 3. We also analyze carefully the first eigenvalues of the Laplacian in the case of the disk with two symmetric cracks placed on a smaller concentric disk in function of their size.

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44

Song, Haiming, and Ran Zhang. "Projection and Contraction Method for the Valuation of American Options." East Asian Journal on Applied Mathematics 5, no. 1 (February 2015): 48–60. http://dx.doi.org/10.4208/eajam.110914.301114a.

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AbstractAn efficient numerical method is proposed for the valuation of American options via the Black-Scholes variational inequality. A far field boundary condition is employed to truncate the unbounded domain problem to produce the bounded domain problem with the associated variational inequality, to which our finite element method is applied. We prove that the matrix involved in the finite element method is symmetric and positive definite, and solve the discretized variational inequality by the projection and contraction method. Numerical experiments are conducted that demonstrate the superior performance of our method, in comparison with earlier methods.

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Vasylyshyn, T. V. "Algebras of symmetric analytic functions on Cartesian powers of Lebesgue integrable in a power $p\in [1,+\infty)$ functions." Carpathian Mathematical Publications 13, no. 2 (August 16, 2021): 340–51. http://dx.doi.org/10.15330/cmp.13.2.340-351.

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The work is devoted to the study of Fréchet algebras of symmetric (invariant under the composition of every of components of its argument with any measure preserving bijection of the domain of components of the argument) analytic functions on Cartesian powers of complex Banach spaces of Lebesgue integrable in a power $p\in [1,+\infty)$ complex-valued functions on the segment $[0,1]$ and on the semi-axis. We show that the Fréchet algebra of all symmetric analytic entire complex-valued functions of bounded type on the $n$th Cartesian power of the complex Banach space $L_p[0,1]$ of all Lebesgue integrable in a power $p\in [1,+\infty)$ complex-valued functions on the segment $[0,1]$ is isomorphic to the Fréchet algebra of all analytic entire functions on $\mathbb C^m,$ where $m$ is the cardinality of the algebraic basis of the algebra of all symmetric continuous complex-valued polynomials on this Cartesian power. The analogical result for the Fréchet algebra of all symmetric analytic entire complex-valued functions of bounded type on the $n$th Cartesian power of the complex Banach space $L_p[0,+\infty)$ of all Lebesgue integrable in a power $p\in [1,+\infty)$ complex-valued functions on the semi-axis $[0,+\infty)$ is proved.

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Li, Keqiang, Shangjiu Wang, and Shaoyong Li. "Symmetry of large solutions for semilinear elliptic equations in a symmetric convex domain." AIMS Mathematics 7, no. 6 (2022): 10860–66. http://dx.doi.org/10.3934/math.2022607.

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<abstract><p>In this paper, we consider the solutions of the boundary blow-up problem</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \begin{cases} \Delta u = \frac{1}{u^\gamma} +f(u) \ \ \ \ \mathrm{in}\ \ \ \Omega,\\ \ u>0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{in}\ \ \ \Omega, \\ \ u = +\infty \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{on} \ \ \partial\Omega, \end{cases} \end{eqnarray*} $\end{document} </tex-math></disp-formula></p> <p>where $ \gamma > 0, \ \Omega $ is a bounded convex smooth domain and symmetric w.r.t. a direction. $ f $ is a locally Lipschitz continuous and non-decreasing function. We prove symmetry and monotonicity of solutions of the problem above by the moving planes method. A maximum principle in narrow domains plays an important role in proof of the main result.</p></abstract>

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DEL PEZZO, LEANDRO M. "OPTIMIZATION PROBLEM FOR EXTREMALS OF THE TRACE INEQUALITY IN DOMAINS WITH HOLES." Communications in Contemporary Mathematics 12, no. 04 (August 2010): 569–86. http://dx.doi.org/10.1142/s0219199710003920.

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We study the Sobolev trace constant for functions defined in a bounded domain Ω that vanish in the subset A. We find a formula for the first variation of the Sobolev trace with respect to the hole. As a consequence of this formula, we prove that when Ω is a centered ball, the symmetric hole is critical when we consider deformation that preserve volume but is not optimal for some case.

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Tordecilla, Jesus Alberto Leon. "Existence of solutions to nonlocal elliptic problems with singular and combined nonlinearities." Electronic Journal of Differential Equations 2022, no. 01-87 (June 27, 2022): 40. http://dx.doi.org/10.58997/ejde.2022.40.

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We use an approximation scheme together with a variation of the fixed point theorem to show the existence of a positive solution to a nonlocal boundary value problem. This problem has a smooth bounded domain in R<sup>N</sup>, a singular term, and combined nonlinearities. We also study the symmetric, monotonicity, and asymptotic behavior of the solutions with respect to a parameter involved in the problem.

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KATAYAMA, SOICHIRO, and HIDEO KUBO. "LOWER BOUND OF THE LIFESPAN OF SOLUTIONS TO SEMILINEAR WAVE EQUATIONS IN AN EXTERIOR DOMAIN." Journal of Hyperbolic Differential Equations 10, no. 02 (June 2013): 199–234. http://dx.doi.org/10.1142/s0219891613500094.

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We consider the Cauchy–Dirichlet problem for semilinear wave equations in a three space-dimensional domain exterior to a bounded and non-trapping obstacle. We obtain a detailed estimate for the lower bound of the lifespan of classical solutions when the size of initial data tends to zero, in a similar spirit to that of the works of John and Hörmander where the Cauchy problem was treated. We show that our estimate is sharp at least for radially symmetric case.

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Helffer, Bernard, Ayman Kachmar, and Nicolas Raymond. "Tunneling for the Robin Laplacian in smooth planar domains." Communications in Contemporary Mathematics 19, no. 01 (November 24, 2016): 1650030. http://dx.doi.org/10.1142/s0219199716500309.

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We study the low-lying eigenvalues of the semiclassical Robin Laplacian in a smooth planar domain with bounded boundary which is symmetric with respect to an axis. In the case when the curvature of the boundary of the domain attains its maximum at exactly two points away from the axis of symmetry, we establish an explicit asymptotic formula for the splitting of the first two eigenvalues. This is a rigorous derivation of the semiclassical tunneling effect induced by the domain’s geometry. Our approach is close to the Born–Oppenheimer one and yields, as a byproduct, a Weyl formula of independent interest.

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