We show that the convergence, as $p\to\infty$, of the solution $u_p$ of the Dirichlet problem for $-\Delta_p u(x)=f(x)$ in a bounded domain $\Omega\subset{\hbox{\bf R}}^n$ with zero-Dirichlet boundary condition and with continuous $f$ in the following cases: (i) one-dimensional case, radial cases; (ii) the case of no balanced family; and (iii) two cases with vanishing integral. We also give some properties of the maximizers for the functional $\int_\Omega f(x)v(x)\d x$ in the space of functions $v\in C(\overline\Omega)\cap W^{1,\infty}(\Omega)$ satisfying $v|_{\partial\Omega}=0$ and $\|Dv\|_{L^\infty(\Omega)}\leq 1$.
Limits of Solutions of p-Laplace Equations as p Goes to Infinity and Related Variational Problems / H., Ishii; Loreti, Paola. - In: SIAM JOURNAL ON MATHEMATICAL ANALYSIS. - ISSN 0036-1410. - STAMPA. - 37:(2005), pp. 411-437. [10.1137/S0036141004432827]
Limits of Solutions of p-Laplace Equations as p Goes to Infinity and Related Variational Problems
LORETI, Paola
2005
Abstract
We show that the convergence, as $p\to\infty$, of the solution $u_p$ of the Dirichlet problem for $-\Delta_p u(x)=f(x)$ in a bounded domain $\Omega\subset{\hbox{\bf R}}^n$ with zero-Dirichlet boundary condition and with continuous $f$ in the following cases: (i) one-dimensional case, radial cases; (ii) the case of no balanced family; and (iii) two cases with vanishing integral. We also give some properties of the maximizers for the functional $\int_\Omega f(x)v(x)\d x$ in the space of functions $v\in C(\overline\Omega)\cap W^{1,\infty}(\Omega)$ satisfying $v|_{\partial\Omega}=0$ and $\|Dv\|_{L^\infty(\Omega)}\leq 1$.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.