Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRegularity and Symmetry for Semilinear Elliptic Equations in Bounded Domains
82
0
0.0
(
0
)
Authors
Louis Dupaigne
Ask ChatGPT about the research
No Arabic abstract
In the present paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are locally stable.
rate research
Read More
We classify stable and finite Morse index solutions to general semilinear elliptic equations posed in Euclidean space of dimension at most 10, or in some unbounded domains.
We prove optimal boundary regularity for bounded positive weak solutions of fast diffusion equations in smooth bounded domains. This solves a problem raised by Berryman and Holland in 1980 for these equations in the subcritical and critical regimes. Our proof of the a priori estimates uses a geometric type structure of the fast diffusion equations, where an important ingredient is an evolution equation for a curvature-like quantity.
In this paper, we consider the pointwise boundary Lipschitz regularity of solutions for the semilinear elliptic equations in divergence form mainly under some weaker assumptions on nonhomogeneous term and the boundary. If the domain satisfies C^{1,text{Dini}} condition at a boundary point, and the nonhomogeneous term satisfies Dini continuous condition and Lipschitz Newtonian potential condition, then the solution is Lipschitz continuous at this point. Furthermore, we generalize this result to Reifenberg C^{1,text{Dini}} domains.
In this paper we prove regularity results for a class of nonlinear degenerate elliptic equations of the form $displaystyle -operatorname{div}(A(| abla u|) abla u)+Bleft( | abla u|right) =f(u)$; in particular, we investigate the second order regularity of the solutions. As a consequence of these results, we obtain symmetry and monotonicity properties of positive solutions for this class of degenerate problems in convex symmetric domains via a suitable adaption of the celebrated moving plane method of Alexandrov-Serrin.
Given a smooth domain $OmegasubsetRR^N$ such that $0 in partialOmega$ and given a nonnegative smooth function $zeta$ on $partialOmega$, we study the behavior near 0 of positive solutions of $-Delta u=u^q$ in $Omega$ such that $u = zeta$ on $partialOmegasetminus{0}$. We prove that if $frac{N+1}{N-1} < q < frac{N+2}{N-2}$, then $u(x)leq C abs{x}^{-frac{2}{q-1}}$ and we compute the limit of $abs{x}^{frac{2}{q-1}} u(x)$ as $x to 0$. We also investigate the case $q= frac{N+1}{N-1}$. The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
