## A nonlocal 1-Laplacian problem and median values.(English)Zbl 1341.45001

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

### MSC:

 45G10 Other nonlinear integral equations 45J05 Integro-ordinary differential equations 47H06 Nonlinear accretive operators, dissipative operators, etc.
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