Let $\Omega$ be a bounded smooth domain in $\rn$, $N\geq 2$, and let us denote by $d(x)=$dist$(x,\partial \Omega)$. We study a class of singular Hamilton-Jacobi equations, arising from stochastic control problems, whose simplest model is $$ - \alpha \Delta u+ u + \frac{\D u \cdot B (x)}{d (x)}+ c(x) |\D u|^2=f (x) \quad \mbox{in } \Omega, $$ where $f$ belongs to $W^{1,\infty}_{\rm loc} (\Omega)$ and is (possibly) singular at $\partial \Omega$, $c\in \lip$ (with no sign condition) and the field $B\in \lip^N$ has the outward direction and satisfies $B\cdot \nu\geq \alpha$ at $\partial \Omega$ ($\nu $ is the outward normal). Despite the singularity in the equation, we prove gradient bounds up to the boundary and the existence of a (globally) Lipschitz solution. In some cases, we show that this is the unique bounded solution. We also discuss the stability of such estimates with respect to $\alpha$, as $\alpha$ vanishes, obtaining Lipschitz solutions for first order problems with similar features. The main tool is a refined weighted version of the classical Bernstein's method to get gradient bounds; the key role is played here by the orthogonal transport component of the Hamiltonian.

Leonori, T., Porretta, A. (2011). Gradient bounds for elliptic problems singular at the boundary. ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 202(2), 663-705 [10.1007/s00205-011-0436-9].

Gradient bounds for elliptic problems singular at the boundary

PORRETTA, ALESSIO
2011-01-01

Abstract

Let $\Omega$ be a bounded smooth domain in $\rn$, $N\geq 2$, and let us denote by $d(x)=$dist$(x,\partial \Omega)$. We study a class of singular Hamilton-Jacobi equations, arising from stochastic control problems, whose simplest model is $$ - \alpha \Delta u+ u + \frac{\D u \cdot B (x)}{d (x)}+ c(x) |\D u|^2=f (x) \quad \mbox{in } \Omega, $$ where $f$ belongs to $W^{1,\infty}_{\rm loc} (\Omega)$ and is (possibly) singular at $\partial \Omega$, $c\in \lip$ (with no sign condition) and the field $B\in \lip^N$ has the outward direction and satisfies $B\cdot \nu\geq \alpha$ at $\partial \Omega$ ($\nu $ is the outward normal). Despite the singularity in the equation, we prove gradient bounds up to the boundary and the existence of a (globally) Lipschitz solution. In some cases, we show that this is the unique bounded solution. We also discuss the stability of such estimates with respect to $\alpha$, as $\alpha$ vanishes, obtaining Lipschitz solutions for first order problems with similar features. The main tool is a refined weighted version of the classical Bernstein's method to get gradient bounds; the key role is played here by the orthogonal transport component of the Hamiltonian.
2011
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - ANALISI MATEMATICA
English
Con Impact Factor ISI
Gary Lieberman, riconosciuto esperto mondiale (oltre 2000 citazioni) della regolarita' di equazioni ellittiche - in particolare regolarita' al bordo - cita il presente lavoro in un suo recente articolo ("Gradient estimates for singular fully nonlinear elliptic equations", 2015) con le seguenti parole: << In this paper, we use [8] as an inspiration for a general nonlinear theory. Some of our techniques are similar to those in [8], but we do not intend to improve (or reprove) all the results in [8]. For example, an important element in [8] is the study of results when aij vanishes on all of Ω. This situation is not studied in the current work, and many of the main results in [8] are also not amenable to our methods. >> Nella sua lista di referenze, [8] si riferisce appunto al presente lavoro con Leonori su ARMA.
http://link.springer.com/article/10.1007%2Fs00205-011-0436-9
Leonori, T., Porretta, A. (2011). Gradient bounds for elliptic problems singular at the boundary. ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 202(2), 663-705 [10.1007/s00205-011-0436-9].
Leonori, T; Porretta, A
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/102221
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