Local and nonlocal weighted \(p\)-Laplacian evolution equations with Neumann boundary conditions. (English) Zbl 1213.35282

The authors studied the existence and uniqueness of solutions of the local diffusion equation
\[ u_t=\text{div}(g|\nabla u|^{p-2}\nabla u)\quad\text{in }\Omega\times (0,T),\;p\geq 1, \]
with Neumann boundary conditions \(g|\nabla u|^{p-2}\nabla u\cdot\eta=0\) on \(\partial\Omega\times (0,T)\) for some function \(g\geq 0\). They proved the existence and uniqueness of the corresponding nonlocal problem
\[ u_t(x,t)=\int_{\Omega}J(x-y)g((x+y)/2)|u(x,t)-u(y,t)|^{p-2}(u(y,t)-u(x,t))\,dy \quad\text{in }\Omega\times (0,T) \] with the function \(g\geq 0\) possibly vanishing on a set of positive measure. Such equation arises in the study of image denoising and reconstruction. The authors also proved the convergence of the nonlocal problem to the local problem for any \(p\geq 1\).


35K92 Quasilinear parabolic equations with \(p\)-Laplacian
47H06 Nonlinear accretive operators, dissipative operators, etc.
35R09 Integro-partial differential equations
35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
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