A class of integral equations and approximation of p-Laplace equations
Tóm tắt
Let
$${\Omega\subset\mathbb{R}^n}$$
be a bounded domain, and let 1 < p < ∞ and σ < p. We study the nonlinear singular integral equation
$$ M[u](x) = f_0(x)\quad {\rm in}\,\Omega$$
with the boundary condition u = g
0 on ∂Ω, where
$${f_0\in C(\overline\Omega)}$$
and
$${g_0\in C(\partial\Omega)}$$
are given functions and M is the singular integral operator given by
$$M[u](x)={\rm p.v.} \int\limits_{B(0,\rho(x))} \frac{p-\sigma}{|z|^{n+\sigma}}|u(x+z)-u(x)|^{p-2} (u(x+z)-u(x))\,{\rm dz},$$
with some choice of
$${\rho\in C(\overline\Omega)}$$
having the property, 0 < ρ(x) ≤ dist (x, ∂Ω). We establish the solvability (well-posedness) of this Dirichlet problem and the convergence uniform on
$${\overline\Omega}$$
, as σ → p, of the solution u
σ
of the Dirichlet problem to the solution u of the Dirichlet problem for the p-Laplace equation νΔ
p
u = f
0 in Ω with the Dirichlet condition u = g
0 on ∂Ω, where the factor ν is a positive constant (see (7.2)).
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