A nonlocal 1-Laplacian problem and median values (Q5963187)

Let \(\Omega\) be a bounded and smooth domain in \(\mathbb{R}^N\), \(u:\Omega\to\mathbb{R}\) a harmonic function, \(J:\mathbb{R}^N\to\mathbb{R}\) be a continuous nonnegative radial function, compactly supported in \(B_1(0)\) with \(J(0)>0\) and NEWLINE\[NEWLINE\int_{\mathbb{R}^n} J(z)\,dz=1,NEWLINE\]NEWLINE a function \(\psi:\Omega_J\to\mathbb{R}\) and denote NEWLINE\[NEWLINE\Omega_J= \Omega+\text{supp\,}J,\quad u_\varphi= u\chi_\Omega+ \psi\chi_{\Omega_J\setminus\overline\Omega},NEWLINE\]NEWLINE and define NEWLINE\[NEWLINE(\forall E\subset B_1(0))\Biggl(\mu^0_J= \int_E J(z)\,dz\Biggr)NEWLINE\]NEWLINE the measure of the set \(E\).NEWLINENEWLINE For a measurable function \(f:\mathbb{R}^n\to \mathbb{R}\), a median value \(m\) of \(f\) with respect to \(\mu^0_J\) is NEWLINE\[NEWLINE\mu^0_J(\{y\in B_1(0); f(y)\geq m\})\geq 2^{-1},\;\mu^0_J(\{y\in B_1(0); f(y)\leq m\})\geq 2^{-1}NEWLINE\]NEWLINE and denote such fact by \(m\in\text{median}_{\mu^0_J}f\).NEWLINENEWLINE Denote by sign the multivalued sign-function defined as NEWLINE\[NEWLINE\text{sign\,}z= \begin{cases} 1,\quad & z>0,\\ [-1,1],\quad & z=0,\\ -1,\quad & z<0.\end{cases}NEWLINE\]NEWLINE The authors study solutions \(u\) to a nonlocal 1-Laplacian equation given by NEWLINE\[NEWLINE\begin{aligned} -\int_{\Omega_J} J(x-y)(u_\psi(y)- u(x))|u_\psi(y)- u(x)|^{-1}dy= 0,\quad & x\in\Omega,\\ u(x)= \psi(x),\quad & x\in\Omega_J\setminus\overline\Omega,\end{aligned}\tag{1}NEWLINE\]NEWLINE with Dirichlet boundary condition.NEWLINENEWLINE If \(\psi\in L^1(\Omega_J\setminus\overline\Omega)\), a function \(u\in L^1(\Omega)\) is calledNEWLINENEWLINE (i) a weak-solution to (1) if NEWLINE\[NEWLINE(\exists g:\Omega_J\times \Omega_J\to \mathbb{R})((g\in L^\infty(\Omega_J\times\Omega_J))\wedge(\| g\|_\infty\leq 1)),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\begin{aligned} J(x-y)g(x,y)\in J(x-y)\text{sign}(u_\psi(y)- u_\psi(x))\quad &\text{a.e. }(x,y)\in\Omega_J\times \Omega_J,\\ -\int_{\Omega_J} J(x-y) g(x,y)\,dy= 0\quad &\text{a.e. }x\in\Omega,\end{aligned}NEWLINE\]NEWLINE (ii) a variatonal solution to (1) if \(u\in L^1(\Omega)\) is a weak-solution to (1) and NEWLINE\[NEWLINEg(x,y)= -g(y,x)\quad\text{a.e. }(x,y)\in \Omega_J\times \Omega_J.NEWLINE\]NEWLINE The authors prove three main theorems.NEWLINENEWLINE Theorem. If \(\psi\in L^1(\Omega_J\setminus\Omega)\) and \(u\in L^1(\Omega)\), the following propositions are equivalent:NEWLINENEWLINE (a) \(u\) is a weak-solution to (1) with Dirichlet datum \(\psi\),NEWLINENEWLINE (b) \(u\) verifies the following nonlocal median value property NEWLINE\[NEWLINE(\forall x\in\Omega)(u(x)= \text{median}_{\mu^0_J}u_\psi(x-\cdot)),NEWLINE\]NEWLINE that is, for all \(x\in\Omega\), NEWLINE\[NEWLINE\mu^x_J(\{y\in B_1(x); u_\psi(y)\geq u(x)\})\geq 2^{-1},\;\mu^x_J(\{y\in B_1(x); u_\psi(y)\leq u(x)\})\geq 2^{-1},NEWLINE\]NEWLINE where NEWLINE\[NEWLINE(\forall E\subset B_1(0))\Biggl(\mu^x_J(E)= \int_E J(x-y)\,dy\Biggr).NEWLINE\]NEWLINE Theorem. If \(\psi\in L^1(\Omega_J\setminus\overline\Omega)\) and \(u\in L^1(\Omega)\), the following propositions are equivalent:NEWLINENEWLINE (c) \(u\) is a variational solution to (1),NEWLINENEWLINE (d) \(u\) is a minimizer of the functional NEWLINE\[NEWLINE{\mathcal J}_\psi(u)= 2^{-1} \int_{\Omega_J}\,\int_{\Omega_J} J(x-y)|u_\psi(y)- u_\psi(x)|\,dx\,dy.NEWLINE\]NEWLINE Theorem. If \(\widetilde\psi\in L^\infty(\partial\Omega)\), \(\psi\in W^{1,1}(\Omega_J\setminus \overline\Omega)\cap L^\infty(\Omega_J\setminus\overline\Omega)\) such that \(\psi|_{\partial\Omega}= \widetilde\psi\), NEWLINE\[NEWLINE(\forall(x,y)\in \Omega_J\times \Omega_J)((|x|\leq|y|))\Rightarrow (J(x)\geq J(y)),NEWLINE\]NEWLINE \(u_\varepsilon\) is a variational solution to (1) for \(J_\varepsilon(x)= \varepsilon^{-(N+1)} J(\varepsilon^{-1} x)\), then, up to a subsequence, \(u_\varepsilon\to u\) in \(L^1(\Omega)\), being \(u\) a solution to NEWLINE\[NEWLINE\begin{aligned} -\text{div}(DU\cdot|Du|^{-1})= 0\quad &\text{in }\Omega,\\ u= h\quad &\text{on }\partial\Omega,\end{aligned}NEWLINE\]NEWLINE with \(h=\widetilde\psi\).