Ancient shrinking spherical interfaces in the Allen-Cahn flow

We consider the parabolic Allen-Cahn equation in $\mathbb{R}^n$, $n\ge 2$, $$u_t= \Delta u + (1-u^2)u \quad \hbox{ in } \mathbb{R}^n \times (-\infty, 0].$$ We construct an ancient radially symmetric solution $u(x,t)$ with any given number $k$ of transition layers between $-1$ and $+1$. At main order they consist of $k$ time-traveling copies of $w$ with spherical interfaces distant $O(\log |t| )$ one to each other as $t\to -\infty$. These interfaces are resemble at main order copies of the {\em shrinking sphere} ancient solution to mean the flow by mean curvature of surfaces: $|x| = \sqrt{- 2(n-1)t}$. More precisely, if $w(s)$ denotes the heteroclinic 1-dimensional solution of $w'' + (1-w^2)w=0$ $w(\pm \infty)= \pm 1$ given by $w(s) = \tanh \left(\frac s{\sqrt{2}} \right)$ we have $$u(x,t) \approx \sum_{j=1}^k (-1)^{j-1}w(|x|-\rho_j(t)) - \frac 12 (1+ (-1)^{k}) \quad \hbox{ as } t\to -\infty$$ where $$\rho_j(t)=\sqrt{-2(n-1)t}+\frac{1}{\sqrt{2}}\left(j-\frac{k+1}{2}\right)\log\left(\frac {|t|}{\log |t| }\right)+ O(1),\quad j=1,\ldots ,k.$

Ancient multiple-layer solutions to the Allen-Cahn equation

We consider the parabolic one-dimensional Allen-Cahn equation $$u_t= u_{xx}+ u(1-u^2)\quad (x,t)\in \mathbb{R}\times (-\infty, 0].$$ The steady state $w(x) =\tanh (x/\sqrt{2})$, connects, as a "transition layer" the stable phases $-1$ and $+1$. We construct a solution $u$ with any given number $k$ of transition layers between $-1$ and $+1$. At main order they consist of $k$ time-traveling copies of $w$ with interfaces diverging one to each other as $t\to -\infty$. More precisely, we find $$u(x,t) \approx \sum_{j=1}^k (-1)^{j-1}w(x-\xi_j(t)) + \frac 12 ((-1)^{k-1}- 1)\quad \hbox{as} t\to -\infty,$$ where the functions $\xi_j(t)$ satisfy a first order Toda-type system. They are given by $$\xi_j(t)=\frac{1}{\sqrt{2}}\left(j-\frac{k+1}{2}\right)\log(-t)+\gamma_{jk},\quad j=1,...,k,$$ for certain explicit constants $\gamma_{jk}.

Boundary singularities of solutions of semilinear elliptic equations with critical Hardy potentials

We study the boundary behaviour of the of (E) -\Gd u-\myfrac{\xk }{d^2(x)}u+g(u)=0, where 0<\xk <\frac{1}{4} and $g$ is a continuous nonndecreasing function in a bounded convex domain of \BBR^N. We first construct the Martin kernel associated to the the linear operator \CL_{\xk }=-\Gd-\frac{\xk }{d^2(x)} and give a general condition for solving equation (E) with any Radon measure \gm for boundary data. When $g(u)=|u|^{q-1}u$ we show the existence of a critical exponent q_c=q_c(N,\xk )>1: when $0<q<q_c$ any measure is eligible for solving (E) with \gm for boundary data; if $q\geq q_c$, a necessary and sufficient condition is expressed in terms of the absolute continuity of \gmwith respect to some Besov capacity. The same capacity characterizes the removable compact boundary sets. At end any positive solution (F) -\Gd u-\frac{\xk }{d^2(x)}u+|u|^{q-1}u=0 with $q>1$ admits a boundary trace which is a positive outer regular Borel measure. When $1<q<q_c$ we prove that to any positive outer regular Borel measure we can associate a positive solutions of ($F$) with this boundary trace.Comment: 77 page

Nonlinear nonlocal equations involving subcritical or power nonlinearities and measure data

Let $s\in(0, 1),$ 1 < p < \frac{N}{s} and $\Omega\subset{\mathbb R}^N$ be an open bounded set. In this work we study the existence of solutions to problems ($E_\pm$) $Lu\pm g(u) = \mu$ and $u = 0$ a.e. in ${\mathbb R}^N\setminus \Omega,$ where $g\in C({\mathbb R})$ is a nondecreasing function, $\mu$ is a bounded Radon measure on $\Omega$ and $L$ is an integro-differential operator with order of differentiability $s\in(0, 1)$ and summability $p\in(1, \frac{N}{s}).$ More precisely, $L$ is a fractional $p$-Laplace type operator. We establish sufficient conditions for the solvability of problems ($E_\pm$). In the particular case $g(t) = |t|^{ \kappa-1}t;$ \kappa > p-1, these conditions are expressed in terms of Bessel capacities

Semilinear elliptic Schr\"odinger equations with singular potentials and absorption terms

Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $\Sigma \subset \Omega$ be a compact, $C^2$ submanifold without boundary, of dimension $k$ with $0\leq k < N-2$. Put $L_\mu = \Delta + \mu d_\Sigma^{-2}$ in $\Omega \setminus \Sigma$, where $d_\Sigma(x) = \mathrm{dist}(x,\Sigma)$ and $\mu$ is a parameter. We investigate the boundary value problem (P) $-L_\mu u + g(u) = \tau$ in $\Omega \setminus \Sigma$ with condition $u=\nu$ on $\partial \Omega \cup \Sigma$, where $g: \mathbb{R} \to \mathbb{R}$ is a nondecreasing, continuous function, and $\tau$ and $\nu$ are positive measures. The complex interplay between the competing effects of the inverse-square potential $d_\Sigma^{-2}$, the absorption term $g(u)$ and the measure data $\tau,\nu$ discloses different scenarios in which problem (P) is solvable. We provide sharp conditions on the growth of $g$ for the existence of solutions. When $g$ is a power function, namely $g(u)=|u|^{p-1}u$ with $p>1$, we show that problem (P) admits several critical exponents in the sense that singular solutions exist in the subcritical cases (i.e. $p$ is smaller than a critical exponent) and singularities are removable in the supercritical cases (i.e. $p$ is greater than a critical exponent). Finally, we establish various necessary and sufficient conditions expressed in terms of appropriate capacities for the solvability of (P).Comment: 40 page

Semilinear elliptic equations involving power nonlinearities and hardy potentials with boundary singularities

Let $\Omega \subset\mathbb{R}^N$ ($N\geq 3$) be a $C^2$ bounded domain and $\Sigma \subset \partial\Omega$ be a $C^2$ compact submanifold without boundary, of dimension $k$, $0\leq k \leq N-1$. We assume that $\Sigma = \{0\}$ if $k = 0$ and $\Sigma=\partial\Omega$ if $k=N-1$. Denote $d_\Sigma(x)=\mathrm{dist}(x,\Sigma)$ and put $L_\mu=\Delta + \mu d_{\Sigma}^{-2}$ where $\mu$ is a parameter. In this paper, we study boundary value problems for equations $-L_\mu u \pm |u|^{p-1}u = 0$ in $\Omega$ with prescribed condition $u=\nu$ on $\partial \Omega$, where $p>1$ and $\nu$ is a given measure on $\partial \Omega$. The nonlinearity $|u|^{p-1}u$ is referred to as \textit{absorption} or \textit{source} depending whether the plus sign or minus sign appears. The distinctive feature of the problems is characterized by the interplay between the concentration of $\Sigma$, the type of nonlinearity, the exponent $p$ and the parameter $\mu$. The absorption case and the source case are sharply different in several aspects and hence require completely different approaches. In each case, we establish various necessary and sufficient conditions expressed in terms of appropriate capacities. In comparison with related works in the literature, by employing a fine analysis, we are able to treat the supercritical ranges for the exponent $p$, and the critical case for the parameter $\mu$, which justifies the novelty of our paper.Comment: 45 pages. arXiv admin note: text overlap with arXiv:2203.0126

Measure boundary value problems for semilinear elliptic equations with critical Hardy potentials

Artículo de publicación ISILet Omega subset of R-N be a bounded C-2 domain and L-kappa = -Delta - kappa/d(2) where d = dist(., partial derivative Omega) and 0 < K <= 1/4. Let alpha(+/-) = 1 +/- root 1 - 4K, lambda(kappa) the first eigenvalue of L-kappa with corresponding positive eigenfunction phi(kappa). If g is a continuous nondecreasing function satisfying integral(infinity)(1) (g(s) + 2_2f vertical bar g(-s)vertical bar)s(-2) (2N-2+alpha/2N-4+alpha+) ds < infinity, then for any Radon measures nu is an element of m(phi kappa) (Omega) and mu is an element of m (partial derivative Omega) there exists a unique weak solution to problem P nu,mu: L(kappa)u + g(u) = nu in Omega, u = mu on partial derivative Omega. If g(r) = vertical bar r vertical bar(q-1) u (q > 1), we prove that, in the supercritical range of q, a necessary and sufficient condition for solving P-0,P-mu with mu > 0 is that mu is absolutely continuous with W respect to the capacity associated with the space B2- (2+alpha+/2q)',(q)'(RN-1). We also characterize the boundary removable sets in terms of this capacity. In the subcritical range of q we classify the isolated singularities of positive solutions.FONDECYT 3140567 MATH-Amsud program 13MATH-03-QUES

Boundary Singularities on a Wedge-like Domain of a Semilinear Elliptic Equation

Artículo de publicación ISILet n >= 2 and let Omega subset of Rn+1 be a Lipschitz wedge-like domain. We construct positive weak solutions of the problem Delta u + u(P) = 0 in Omega that vanish in a suitable trace sense on partial derivative Omega, but which are singular at a prescribed 'edge' of Omega if p is equal to or slightly above a certain exponent p(0) > 1 that depends on Omega. Moreover, for the case in which Omega is unbounded, the solutions have fast decay at infinity