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We consider a homogeneous elliptic Dirichlet problem involving an Ornstein-Uhlenbeck operator in a half space $R_{+}^{2}$ of $R^{2}$ . We show that for a particular initial datum, which is Lipschitz continuous and bounded on $R_{+}^{2}$ , the second derivative of the classical solution is not uniformly continuous on $R_{+}^{2}$ . In particular this implies that the well known maximal Hölder-regularity results fail in general for Dirichlet problems in unbounded domains involving unbounded coefficients.

A Counter-Example to the Boundary Regularity of Solutions to Elliptic Quasilinear Systems.

Mariano Giaquinta (1978)

Manuscripta mathematica

A counterexample to the “hot spots" conjecture.

Burdzy, Krzysztof, Werner, Wendelin (1999)

Annals of Mathematics. Second Series

A criterion of Petrowsky's kind for a degenerate quasilinear parabolic equation.

Peter Lindqvist (1995)

Revista Matemática Iberoamericana

The celebrated criterion of Petrowsky for the regularity of the latest boundary point, originally formulated for the heat equation, is extended to the so-called p-parabolic equation. A barrier is constructed by the aid of the Barenblatt solution.

A critical growth rate for harmonic and subharmonic functions in the open ball in $R^{n}$

Krzysztof Samotij (1987)

Colloquium Mathematicae

A decay estimate for a class of hyperbolic pseudo-differential equations

Sandra Lucente, Guido Ziliotti (1999)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We consider the equation $u_{t} - i Λ u = 0$ , where $Λ = λ (D_{x})$ is a first order pseudo-differential operator with real symbol $λ (ξ)$ . Under a suitable convexity assumption on $λ$ we find the decay properties for $u (t, x)$ . These can be applied to the linear Maxwell system in anisotropic media and to the nonlinear Cauchy Problem $u_{t} - i Λ u = f (u)$ , $u (0, x) = g (x)$ . If $f (u)$ is a smooth function which satisfies $f (u) ≃ {|u|}^{p}$ near $u = 0$ , and $g$ is small in suitably Sobolev norm, we prove global existence theorems provided $p$ is greater than a critical exponent.

A degenerate parabolic equation in noncylindrical domains.

M. Bertsch, R. Dal Passo, B. Franchi (1992)

Mathematische Annalen

A description of blow-up for the solid fuel ignition model

Bebernes, Jerrold W. (1986)

Equadiff 6

A description of self-similar blow-up for dimensions $n \geq 3$

J. Bebernes, D. Eberly (1988)

Annales de l'I.H.P. Analyse non linéaire

A diffused interface whose chemical potential lies in a Sobolev space

Yoshihiro Tonegawa (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We study a singular perturbation problem arising in the scalar two-phase field model. Given a sequence of functions with a uniform bound on the surface energy, assume the Sobolev norms $W^{1, p}$ of the associated chemical potential fields are bounded uniformly, where $p > \frac{n}{2}$ and $n$ is the dimension of the domain. We show that the limit interface as $ε$ tends to zero is an integral varifold with a sharp integrability condition on the mean curvature.

A diffusion equation for composite materials.

El Hajji, Mohamed (2000)

Electronic Journal of Differential Equations (EJDE) [electronic only]

A Dirichlet problem in the strip.

Montefusco, Eugenio (1996)

Electronic Journal of Differential Equations (EJDE) [electronic only]

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