Let $1le pleinfty$. We show that a function $uin C(mathbb R^N)$ is a viscosity solution to the normalized $p$-Laplace equation $Delta_p^n u(x)=0$ if and only if the asymptotic formula $$ u(x)=mu_p(e,u)(x)+o(e^2) $$ holds as $e o 0$ in the viscosity sense. Here, $mu_p(e,u)(x)$ is the $p$-mean value of $u$ on $B_e(x)$ characterized as a unique minimizer of $$ r u-la r_{L^p(B_e(x))} $$ with respect to $lambdainBbb R$. This kind of asymptotic mean value property (AMVP) extends to the case $p=1$ previous (AMVP)'s obtained when $mu_p(e,u)(x)$ is replaced by other kinds of mean values. The natural definition of $mu_p(e,u)(x)$ makes sure that this is a monotonic and continuous (in the appropriate topology) functional of $u$. These two properties help to establish a fairly general proof of (AMVP), that can also be extended to the (normalized) parabolic $p$-Laplace equation.
A natural approach to the asymptotic mean value property for the p-Laplacian / Ishiwata, Michinori; Magnanini, Rolando; Wadade, Hidemitsu. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - STAMPA. - 56:(2017), pp. 97.1-97.22. [10.1007/s00526-017-1188-7]
A natural approach to the asymptotic mean value property for the p-Laplacian
MAGNANINI, ROLANDO;
2017
Abstract
Let $1le pleinfty$. We show that a function $uin C(mathbb R^N)$ is a viscosity solution to the normalized $p$-Laplace equation $Delta_p^n u(x)=0$ if and only if the asymptotic formula $$ u(x)=mu_p(e,u)(x)+o(e^2) $$ holds as $e o 0$ in the viscosity sense. Here, $mu_p(e,u)(x)$ is the $p$-mean value of $u$ on $B_e(x)$ characterized as a unique minimizer of $$ r u-la r_{L^p(B_e(x))} $$ with respect to $lambdainBbb R$. This kind of asymptotic mean value property (AMVP) extends to the case $p=1$ previous (AMVP)'s obtained when $mu_p(e,u)(x)$ is replaced by other kinds of mean values. The natural definition of $mu_p(e,u)(x)$ makes sure that this is a monotonic and continuous (in the appropriate topology) functional of $u$. These two properties help to establish a fairly general proof of (AMVP), that can also be extended to the (normalized) parabolic $p$-Laplace equation.File | Dimensione | Formato | |
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