Academic literature on the topic 'Almgren monotonicity formula'

AbstractWe study a Schrödinger system of four equations with linear coupling functions and nonlinear couplings, including the case that the corresponding elliptic operators are indefinite. For any given nonlinear coupling {\beta>0}, we first use minimizing sequences on a normalized set to obtain a minimizer, which implies the existence of positive solutions for some linear coupling constants {\mu_{\beta},\nu_{\beta}} by Lagrange multiplier rules. Then, as {\beta\to\infty}, we prove that the limit configurations to the competing system are segregated in two groups, develop a variant of Almgren’s monotonicity formula to reveal the Lipschitz continuity of the limit profiles and establish a kind of local Pohozaev identity to obtain the extremality conditions. Finally, we study the relation between the limit profiles and the optimal partition for principal eigenvalue of the elliptic system and obtain an optimal partition for principal eigenvalues of elliptic systems.

In this note, we prove two monotonicity formulas for solutions of $\Delta _H f = c$ and $\Delta _H f - \partial _t f = c$ in Carnot groups. Such formulas involve the right-invariant carré du champ of a function and they are false for the left-invariant one. The main results, theorems 1.1 and 1.2, display a resemblance with two deep monotonicity formulas respectively due to Alt–Caffarelli–Friedman for the standard Laplacian, and to Caffarelli for the heat equation. In connection with this aspect we ask the question whether an ‘almost monotonicity’ formula be possible. In the last section, we discuss the failure of the nondecreasing monotonicity of an Almgren type functional.

AbstractWe study the singular set in the thin obstacle problem for degenerate parabolic equations with weight $$|y|^a$$ | y | a for $$a \in (-1,1)$$ a ∈ ( - 1 , 1 ) . Such problem arises as the local extension of the obstacle problem for the fractional heat operator $$(\partial _t - \Delta _x)^s$$ ( ∂ t - Δ x ) s for $$s \in (0,1)$$ s ∈ ( 0 , 1 ) . Our main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve it, we prove Almgren-Poon, Weiss, and Monneau type monotonicity formulas which generalize those for the case of the heat equation ($$a=0$$ a = 0 ).

AbstractWe study the nodal set of stationary solutions to equations of the form $$(-\Delta )^s u = \lambda _+ (u_+)^{q-1} - \lambda _- (u_-)^{q-1}\quad \text {in }B_1,$$ ( - Δ ) s u = λ + ( u + ) q - 1 - λ - ( u - ) q - 1 in B 1 , where $$\lambda _+,\lambda _->0, q \in [1,2)$$ λ + , λ - > 0 , q ∈ [ 1 , 2 ) , and $$u_+$$ u + and $$u_-$$ u - are respectively the positive and negative part of u. This collection of nonlinearities includes the unstable two-phase membrane problem $$q=1$$ q = 1 as well as sublinear equations for $$1<q<2$$ 1 < q < 2 . We initially prove the validity of the strong unique continuation property and the finiteness of the vanishing order, in order to implement a blow-up analysis of the nodal set. As in the local case $$s=1$$ s = 1 , we prove that the admissible vanishing orders can not exceed the critical value $$k_q= 2s/(2- q)$$ k q = 2 s / ( 2 - q ) . Moreover, we study the regularity of the nodal set and we prove a stratification result. Ultimately, for those parameters such that $$k_q< 1$$ k q < 1 , we prove a remarkable difference with the local case: solutions can only vanish with order $$k_q$$ k q and the problem admits one dimensional solutions. Our approach is based on the validity of either a family of Almgren-type or a 2-parameter family of Weiss-type monotonicity formulas, according to the vanishing order of the solution.

This thesis is devoted to the study of several problems arising in the field of nonlinear analysis. The work is divided in two parts: the first one concerns existence of oscillating solutions, in a suitable sense, for some nonlinear ODEs and PDEs, while the second one regards the study of qualitative properties, such as monotonicity and symmetry, for solutions to some elliptic problems in unbounded domains. Although the topics faced in this work can appear far away one from the other, the techniques employed in different chapters share several common features. In the firts part, the variational structure of the considered problems plays an essential role, and in particular we obtain existence of oscillating solutions by means of non-standard versions of the Nehari's method and of the Seifert's broken geodesics argument. In the second part, classical tools of geometric analysis, such as the moving planes method and the application of Liouville-type theorems, are used to prove 1-dimensional symmetry of solutions in different situations.