Related papers: Symmetry results for cooperative elliptic systems …
We show that every entire solution to the Bernoulli (or one-phase) free boundary problem with finite Morse index in $\mathbb{R}^3$ is axially symmetric. In fact, we additionally prove that the same result would follow in any dimension $4…
We provide symmetrization results in the form of mass concentration comparisons for fractional singular elliptic equations in bounded domains, coupled with homogeneous external Dirichlet conditions. Two types of comparison results are…
We prove a sharp Schwarz-type lemma for meromorphic functions with spherical derivative uniformly bounded away from zero. As a consequence we deduce an improved quantitative version of a recent normality criterion due to Grahl & Nevo and…
We establish a positivity property for a class of semilinear elliptic problems involving indefinite sublinear nonlinearities. Namely, we show that any nontrivial nonnegative solution is positive for a class of problems the strong maximum…
This paper is concerned with a class of nonhomogeneous quasilinear elliptic system driven by the locally symmetric potential and the small continuous perturbations in the whole-space $\mathbb{R}^N$. By a variant of Clark's theorem without…
The purpose of this paper is to study properties of continua (closed connected sets) of nontrivial solutions of non-cooperative elliptic systems considered on geodesic balls in $S^n$. In particular, we have shown that if the geodesic ball…
Nonlinear ODEs invariant under the group SL(2,R) are solved numerically. We show that solution methods incorporating the Lie point symmetries provide better results than standard methods.
Over the last two decades, classical Schwarz methods have been extended to systems of hyperbolic partial differential equations, and it was observed that the classical Schwarz method can be convergent even without overlap in certain cases.…
We consider a strongly nonlinear elliptic problem with the homogeneous Dirichlet boundary condition. The growth and the coercivity of the elliptic operator is assumed to be indicated by an inhomogeneous anisotropic $\mathcal{N}$-function.…
In this paper we complement the program concerning the application of symmetrization methods to nonlocal PDEs by providing new estimates, in the sense of mass concentration comparison, for solutions to linear fractional elliptic and…
In this paper, we consider radial symmetry property of positive solutions of an integral equation arising from some higher order semi-linear elliptic equations on the whole space $\mathbf{R}^n$. We do not use the usual way to get symmetric…
We show, using symmetrization techniques, that it is possible to prove a comparison principle (we are mainly focused on $L^1$ comparison) between solutions to an elliptic partial differential equation on a smooth bounded set $\Omega$ with a…
In this paper, we study the existence of nontrivial solutions of the Dirichlet boundary value problem for the following elliptic system: \begin{equation} \left\{ \begin{aligned} -\Delta u & = au + bv + f(x,u,v); &\quad\mbox{ for…
Smooth axially symmetric Helfrich topological spheres are either round or else they must satisfy a second order equation known as the reduced membrane equation [17]. In this paper, we show that, conversely, axially symmetric closed genus…
In this paper we prove classification results for solutions to subcritical and critical semilinear elliptic equations with a nonnegative potential on noncompact manifolds with nonnegative Ricci curvature. We show in the subcritical case…
Symmetry groups allow to transform solutions of differential equations continuously into other solutions. This property can be used for the observability analysis of infinite-dimensional systems with input and output. In this contribution,…
We consider positive solutions to a singular semilinear elliptic equation in bounded smooth domains, with zero Dirichlet boundary conditions. We provide some weak and strong maximum principles for the H^1_0 part of the solution that allow…
We extend the results of Schapira and Schneiders on relative regularity and finiteness of elliptic pairs to the framework of $\shd[[\hbar]]$-modules and $\R$-constructible sheaves of $\C[[\h]]$-modules. We also construct a relative duality…
Comparison results for solutions to the Dirichlet problems for a class of nonlinear, anisotropic parabolic equations are established. These results are obtained through a semi-discretization method in time after providing estimates for…
The issue of symmetry and symmetry breaking is fundamental in all areas of science. Symmetry is often assimilated to order and beauty while symmetry breaking is the source of many interesting phenomena such as phase transitions,…