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Journal articles on the topic 'Fully-nonlinear elliptic PDE'

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Author: Grafiati

Published: 4 June 2021

Last updated: 31 January 2023

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1

Sirakov, Boyan. "Solvability of Uniformly Elliptic Fully Nonlinear PDE." Archive for Rational Mechanics and Analysis 195, no. 2 (May 6, 2009): 579–607. http://dx.doi.org/10.1007/s00205-009-0218-9.

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2

Ikoma, Norihisa, and Hitoshi Ishii. "Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 29, no. 5 (September 2012): 783–812. http://dx.doi.org/10.1016/j.anihpc.2012.04.004.

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3

KOIKE, Shigeaki, and Andrzej ŚWIĘCH. "Weak Harnack inequality for fully nonlinear uniformly elliptic PDE with unbounded ingredients." Journal of the Mathematical Society of Japan 61, no. 3 (July 2009): 723–55. http://dx.doi.org/10.2969/jmsj/06130723.

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4

Ikoma, Norihisa, and Hitoshi Ishii. "Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls, II." Bulletin of Mathematical Sciences 5, no. 3 (July 25, 2015): 451–510. http://dx.doi.org/10.1007/s13373-015-0071-0.

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5

Ayanbayev, Birzhan, and Nikos Katzourakis. "On the Inverse Source Identification Problem in $L^{\infty }$ for Fully Nonlinear Elliptic PDE." Vietnam Journal of Mathematics 49, no. 3 (July 22, 2021): 815–29. http://dx.doi.org/10.1007/s10013-021-00515-6.

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AbstractIn this paper we generalise the results proved in N. Katzourakis (SIAM J. Math. Anal. 51, 1349–1370, 2019) by studying the ill-posed problem of identifying the source of a fully nonlinear elliptic equation. We assume Dirichlet data and some partial noisy information for the solution on a compact set through a fully nonlinear observation operator. We deal with the highly nonlinear nonconvex nature of the problem and the lack of weak continuity by introducing a two-parameter Tykhonov regularisation with a higher order L2 “viscosity term” for the $L^{\infty }$ L ∞ minimisation problem which allows to approximate by weakly lower semicontinuous cost functionals.

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6

Barles, G., and Jérôme Busca. "EXISTENCE AND COMPARISON RESULTS FOR FULLY NONLINEAR DEGENERATE ELLIPTIC EQUATIONS WITHOUT ZEROTH-ORDER TERM1*." Communications in Partial Differential Equations 26, no. 11-12 (November 1, 2001): 2323–37. http://dx.doi.org/10.1081/pde-100107824.

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7

Jensen, Robert, and Andrzej Świech. "Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE." Communications on Pure & Applied Analysis 4, no. 1 (2005): 199–207. http://dx.doi.org/10.3934/cpaa.2005.4.187.

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8

Caffarelli, Luis A., and Panagiotis E. Souganidis. "Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media." Inventiones mathematicae 180, no. 2 (January 8, 2010): 301–60. http://dx.doi.org/10.1007/s00222-009-0230-6.

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9

Philippin, G. A., and A. Safoui. "Some applications of the maximum principle to a variety of fully nonlinear elliptic PDE?s." Zeitschrift f�r Angewandte Mathematik und Physik (ZAMP) 54, no. 5 (September 1, 2003): 739–55. http://dx.doi.org/10.1007/s00033-003-3200-7.

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10

Katzourakis, Nikos. "On linear degenerate elliptic PDE systems with constant coefficients." Advances in Calculus of Variations 9, no. 3 (July 1, 2016): 283–91. http://dx.doi.org/10.1515/acv-2015-0004.

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AbstractLet ${\mathbf{A}}$ be a symmetric convex quadratic form on ${\mathbb{R}^{Nn}}$ and Ω $\subset$$\mathbb{R}^{n}$ a bounded convex domain. We consider the problem of existence of solutions u: Ω $\subset$$\mathbb{R}^{n}$$\to$$\mathbb{R}^{N}$ to the problem${}\left\{\begin{aligned} \displaystyle\sum_{\beta=1}^{N}\sum_{i,j=1}^{n}% \mathbf{A}_{\alpha i\beta j}D^{2}_{ij}u_{\beta}&\displaystyle=f_{\alpha}&&% \displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\right.\phantom{\}}$when ${f\in L^{2}(\Omega,\mathbb{R}^{N})}$. Problem (1) is degenerate elliptic and it has not been considered before without the assumption of strict rank-one convexity. In general, it may not have even distributional solutions. By introducing an extension of distributions adapted to (1), we prove existence, partial regularity and by imposing an extra condition uniqueness as well. The satisfaction of the boundary condition is also an issue due to the low regularity of the solution. The motivation to study (1) and the method of the proof arose from recent work of the author [10] on generalised solutions for fully nonlinear systems.

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11

Katzourakis, Nikos. "Weak vs. 𝒟-solutions to linear hyperbolic first-order systems with constant coefficients." Journal of Hyperbolic Differential Equations 15, no. 02 (June 2018): 329–47. http://dx.doi.org/10.1142/s0219891618500121.

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We establish a consistency result by comparing two independent notions of generalized solutions to a large class of linear hyperbolic first-order PDE systems with constant coefficients, showing that they eventually coincide. The first is the usual notion of weak solutions defined via duality. The second is the new notion of [Formula: see text]-solutions which we recently introduced and arose in connection to the vectorial calculus of variations in [Formula: see text] and fully nonlinear elliptic systems. This new approach is a duality-free alternative to distributions and is based on the probabilistic representation of limits of difference quotients.

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12

Wang, Pei-Yong. "REGULARITY OF FREE BOUNDARIES OF TWO-PHASE PROBLEMS FOR FULLY NONLINEAR ELLIPTIC EQUATIONS OF SECOND ORDER. II. FLAT FREE BOUNDARIES ARE LIPSCHITZ." Communications in Partial Differential Equations 27, no. 7-8 (January 7, 2002): 1497–514. http://dx.doi.org/10.1081/pde-120005846.

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13

Kraus, Johannes, Svetoslav Nakov, and Sergey I. Repin. "Reliable Numerical Solution of a Class of Nonlinear Elliptic Problems Generated by the Poisson–Boltzmann Equation." Computational Methods in Applied Mathematics 20, no. 2 (April 1, 2020): 293–319. http://dx.doi.org/10.1515/cmam-2018-0252.

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AbstractWe consider a class of nonlinear elliptic problems associated with models in biophysics, which are described by the Poisson–Boltzmann equation (PBE). We prove mathematical correctness of the problem, study a suitable class of approximations, and deduce guaranteed and fully computable bounds of approximation errors. The latter goal is achieved by means of the approach suggested in [19] for convex variational problems. Moreover, we establish the error identity, which defines the error measure natural for the considered class of problems and show that it yields computable majorants and minorants of the global error as well as indicators of local errors that provide efficient adaptation of meshes. Theoretical results are confirmed by a collection of numerical tests that includes problems on 2D and 3D Lipschitz domains.

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14

Święch, Andrzej. "A note on the upper perturbation property and removable sets for fully nonlinear degenerate elliptic PDE." Nonlinear Differential Equations and Applications NoDEA 26, no. 1 (January 3, 2019). http://dx.doi.org/10.1007/s00030-018-0547-1.

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15

Bueno, Antonio, and Irene Ortiz. "Surfaces of prescribed linear Weingarten curvature in." Proceedings of the Royal Society of Edinburgh: Section A Mathematics, July 22, 2022, 1–24. http://dx.doi.org/10.1017/prm.2022.48.

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Given $a,\,b\in \mathbb {R}$ and $\Phi \in C^{1}(\mathbb {S}^{2})$ , we study immersed oriented surfaces $\Sigma$ in the Euclidean 3-space $\mathbb {R}^{3}$ whose mean curvature $H$ and Gauss curvature $K$ satisfy $2aH+bK=\Phi (N)$ , where $N:\Sigma \rightarrow \mathbb {S}^{2}$ is the Gauss map. This theory widely generalizes some of paramount importance such as the ones constant mean and Gauss curvature surfaces, linear Weingarten surfaces and self-translating solitons of the mean curvature flow. Under mild assumptions on the prescribed function $\Phi$ , we exhibit a classification result for rotational surfaces in the case that the underlying fully nonlinear PDE that governs these surfaces is elliptic or hyperbolic.

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