Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userComparison results for semilinear elliptic equations using a new symmetrization method
541
0
0.0
(
0
)
Authors
Franc{c}ois Hamel
Ask ChatGPT about the research
No Arabic abstract
In this paper, we prove some pointwise comparison results between the solutions of some second-order semilinear elliptic equations in a domain $Omega$ of $R^n$ and the solutions of some radially symmetric equations in the equimeasurable ball $Omega^*$. The coefficients of the symmetrized equations in~$Omega^*$ satisfy similar constraints as the original ones in~$Omega$. We consider both the case of equations with linear growth in the gradient and the case of equations with at most quadratic growth in the gradient. Lastly, we show some improved quantified comparisons when the original domain is not a ball. The method is based on a symmetrization of the second-order terms.
rate research
Read More
We study the existence and multiplicity of nonnegative solutions, as well as the behaviour of corresponding parameter-dependent branches, to the equation $-Delta u = (1-u) u^m - lambda u^n$ in a bounded domain $Omega subset mathbb{R}^N$ endowed with the zero Dirichlet boundary data, where $0<m leq 1$ and $n>0$. When $lambda > 0$, the obtained solutions can be seen as steady states of the corresponding reaction-diffusion equation describing a model of isothermal autocatalytic chemical reaction with termination. In addition to the main new results, we formulate a few relevant conjectures.
A characterization of a semilinear elliptic partial differential equation (PDE) on a bounded domain in $mathbb{R}^n$ is given in terms of an infinite-dimensional dynamical system. The dynamical system is on the space of boundary data for the PDE. This is a novel approach to elliptic problems that enables the use of dynamical systems tools in studying the corresponding PDE. The dynamical system is ill-posed, meaning solutions do not exist forwards or backwards in time for generic initial data. We offer a framework in which this ill-posed system can be analyzed. This can be viewed as generalizing the theory of spatial dynamics, which applies to the case of an infinite cylindrical domain.
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.
In the present paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are locally stable.
Let $Omega subset {mathbb R}^N$ ($N geq 3$) be a $C^2$ bounded domain and $delta$ be the distance to $partial Omega$. We study positive solutions of equation (E) $-L_mu u+ g(| abla u|) = 0$ in $Omega$ where $L_mu=Delta + frac{mu}{delta^2} $, $mu in (0,frac{1}{4}]$ and $g$ is a continuous, nondecreasing function on ${mathbb R}_+$. We prove that if $g$ satisfies a singular integral condition then there exists a unique solution of (E) with a prescribed boundary datum $ u$. When $g(t)=t^q$ with $q in (1,2)$, we show that equation (E) admits a critical exponent $q_mu$ (depending only on $N$ and $mu$). In the subcritical case, namely $1<q<q_mu$, we establish some a priori estimates and provide a description of solutions with an isolated singularity on $partial Omega$. In the supercritical case, i.e. $q_muleq q<2$, we demonstrate a removability result in terms of Bessel capacities.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
