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Discrete & Continuous Dynamical Systems - S

2011 , Volume 4 , Issue 4

Issue on geometric properties for parabolic and elliptic PDE's

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Preface
Filippo Gazzola, Rolando Magnanini and Shigeru Sakaguchi
2011, 4(4): i-ii doi: 10.3934/dcdss.2011.4.4i +[Abstract](348) +[PDF](93.0KB)
Abstract:
Qualitative aspects of parabolic and elliptic partial differential equations have attracted much attention from the early beginnings. In recent years, once basic issues about PDE's, such as existence, uniqueness, stability and regularity of solutions of initial/boundary value problems, have been quite understood, research on topological and/or geometric properties of their solutions have become more intense.

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Symmetries in an overdetermined problem for the Green's function
Virginia Agostiniani and Rolando Magnanini
2011, 4(4): 791-800 doi: 10.3934/dcdss.2011.4.791 +[Abstract](650) +[PDF](330.8KB)
Abstract:
We consider in the plane the problem of reconstructing a domain from the normal derivative of its Green's function with pole at a fixed point in the domain. By means of the theory of conformal mappings, we obtain existence, uniqueness, (non-spherical) symmetry results, and a formula relating the curvature of the boundary of the domain to the normal derivative of its Green's function.
A remark on Hardy type inequalities with remainder terms
Angelo Alvino, Roberta Volpicelli and Bruno Volzone
2011, 4(4): 801-807 doi: 10.3934/dcdss.2011.4.801 +[Abstract](557) +[PDF](337.1KB)
Abstract:
In this paper we focus our attention to some Hardy type inequalities with a remainder term. In particular we find the best value of the constant $h$ for the inequalities

$\int_{\Omega}|\nabla u|^2 dx \geq c \int_{\Omega}\frac{u^2}{|x|^2} dx+ h\int_{\Omega}\frac{u^2}{|x|}dx, \forall u\in H_0^1( \Omega) $

$ \int_{\Omega}|\nabla u|^2dx\geq c\int_{\Omega} \frac{u^2}{|x|^2}dx+ h(\int_{\Omega}|\nabla u| dx)^2, \forall u\in H_0^1 (\Omega)$

where $c\geq 0$ is smaller than the optimal Hardy constant $(N-2)^2/4$.

Positive solutions to a linearly perturbed critical growth biharmonic problem
Elvise Berchio and Filippo Gazzola
2011, 4(4): 809-823 doi: 10.3934/dcdss.2011.4.809 +[Abstract](480) +[PDF](409.1KB)
Abstract:
Existence and nonexistence results for positive solutions to a linearly perturbed critical growth biharmonic problem under Steklov boundary conditions, are determined. Furthermore, by investigating the critical dimensions for this problem, a Sobolev inequality with remainder terms, of both interior and boundary type, is deduced.
Shape optimization for Monge-Ampère equations via domain derivative
Barbara Brandolini, Carlo Nitsch and Cristina Trombetti
2011, 4(4): 825-831 doi: 10.3934/dcdss.2011.4.825 +[Abstract](484) +[PDF](318.4KB)
Abstract:
In this note we prove that, if $\Omega$ is a smooth, strictly convex, open set in $R^n$ $(n \ge 2)$ with given measure, the $L^1$ norm of the convex solution to the Dirichlet problem $\det D^2 u=1$ in $\Omega$, $u=0$ on $\partial\Omega$, is minimum whenever $\Omega$ is an ellipsoid.
Hot spots for the two dimensional heat equation with a rapidly decaying negative potential
Kazuhiro Ishige and Y. Kabeya
2011, 4(4): 833-849 doi: 10.3934/dcdss.2011.4.833 +[Abstract](435) +[PDF](439.7KB)
Abstract:
We consider the Cauchy problem of the two dimensional heat equation with a radially symmetric, negative potential $-V$ which behaves like $V(r)=O(r^{-\kappa})$ as $r\to\infty$, for some $\kappa > 2$. We study the rate and the direction for hot spots to tend to the spatial infinity. Furthermore we give a sufficient condition for hot spots to consist of only one point for any sufficiently large $t>0$.
On a new kind of convexity for solutions of parabolic problems
Kazuhiro Ishige and Paolo Salani
2011, 4(4): 851-864 doi: 10.3934/dcdss.2011.4.851 +[Abstract](576) +[PDF](401.7KB)
Abstract:
We introduce the notion of $\alpha$-parabolic quasi-concavity for functions of space and time, which extends the usual notion of quasi-concavity and the notion of parabolic quasi-cocavity introduced in [18]. Then we investigate the $\alpha$-parabolic quasi-concavity of solutions to parabolic problems with vanishing initial datum. The results here obtained are generalizations of some of the results of [18].
On the motion of polygonal curves with asymptotic lines by crystalline curvature flow with bulk effect
Tetsuya Ishiwata
2011, 4(4): 865-873 doi: 10.3934/dcdss.2011.4.865 +[Abstract](490) +[PDF](314.1KB)
Abstract:
The behavior of polygonal curves with asymptotic lines to crystalline motion with the bulk effect is discussed. We show sufficient conditions for global existence of the solutions and characterize facet-extinction patterns. We also show the eventual monotonicity of shape of the solution curves, that is, the solutions become V-shaped in finite time.
The degenerate drift-diffusion system with the Sobolev critical exponent
T. Ogawa
2011, 4(4): 875-886 doi: 10.3934/dcdss.2011.4.875 +[Abstract](553) +[PDF](407.6KB)
Abstract:
We consider the drift-diffusion system of degenerated type. For $n\ge 3$,

$\partial_t \rho -\Delta \rho^\alpha + \kappa\nabla\cdot (\rho \nabla \psi ) =0, t>0, x \in R^n,$

$-\Delta \psi = \rho, t>0, x \in R^n,$

$\rho(0,x) = \rho_0(x)\ge 0, x \in R^n,$

where $\alpha>1$ and $\kappa=1$. There exists a critical exponent that classifies the global behavior of the weak solution. In particular, we consider the critical case $\alpha_*=\frac{2 n}{n+2}=(2^*)'$, where the Talenti function $U(x)$ solving $-2^*\Delta U^{\frac{n-2}{n+2}}=U$ in $R^n$ classifies the global existence of the weak solution and finite blow-up of the solution.

A Liouville-type theorem for some Weingarten hypersurfaces
Shigeru Sakaguchi
2011, 4(4): 887-895 doi: 10.3934/dcdss.2011.4.887 +[Abstract](602) +[PDF](356.3KB)
Abstract:
We consider the entire graph $G$ of a globally Lipschitz continuous function $u$ over $R^N$ with $N \ge 2$, and consider a class of some Weingarten hypersurfaces in $R^{N+1}$. It is shown that, if $u$ solves in the viscosity sense in $R^N$ the fully nonlinear elliptic equation of a Weingarten hypersurface belonging to this class, then $u$ is an affine function and $G$ is a hyperplane. This result is regarded as a Liouville-type theorem for a class of fully nonlinear elliptic equations. The special case for some Monge-Ampère-type equation is related to the previous result of Magnanini and Sakaguchi which gave some characterizations of the hyperplane by making use of stationary isothermic surfaces.
Singular backward self-similar solutions of a semilinear parabolic equation
Shota Sato and Eiji Yanagida
2011, 4(4): 897-906 doi: 10.3934/dcdss.2011.4.897 +[Abstract](580) +[PDF](318.3KB)
Abstract:
We consider a parabolic partial differential equation with power nonlinearity. Our concern is the existence of a singular solution whose singularity becomes anomalous in finite time. First we study the structure of singular radial solutions for an equation derived by backward self-similar variables. Using this, we obtain a singular backward self-similar solution whose singularity becomes stronger or weaker than that of a singular steady state.
An eigenvalue problem related to blowing-up solutions for a semilinear elliptic equation with the critical Sobolev exponent
Futoshi Takahashi
2011, 4(4): 907-922 doi: 10.3934/dcdss.2011.4.907 +[Abstract](516) +[PDF](425.1KB)
Abstract:
We consider the eigenvalue problem

$ -\Delta v = \lambda ( c_0 p u^{p-1}_\varepsilon + \varepsilon) v$ in $\Omega,$

$ v = 0$ on $\partial\Omega,$

$ || v ||_{L^\infty(\Omega)} = 1$

where $\Omega \subset R^N (N \ge 5)$ is a smooth bounded domain, $c_0 = N(N-2)$, $p = (N+2)/(N-2)$ is the critical Sobolev exponent and $\varepsilon >0$ is a small parameter. Here $u_\varepsilon $ is a positive solution of

$ -\Delta u = c_0 u^p + \varepsilon u $ in $ \Omega, \quad u|_{\partial \Omega} = 0 $

with the property that

$ \frac{\int_\Omega |\nabla u_\varepsilon |^2 dx} {( \int_\Omega |u_\varepsilon |^{p+1} dx )^{\frac{2}{p+1}}} \to S_N$ as $\varepsilon\to 0, $

where $S_N$ is the best constant for the Sobolev inequality. In this paper, we show several asymptotic estimates for the eigenvalues $\lambda_{i, \varepsilon}$ and corresponding eigenfunctions $v_{i,\varepsilon}$ for $i=1, 2, \cdots, N+1, N+2$.

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