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Geometric Properties for Parabolic and Elliptic PDE's

herausgegeben von: Prof. Vincenzo Ferone, Prof. Tatsuki Kawakami, Prof. Paolo Salani, Prof. Futoshi Takahashi

Verlag: Springer International Publishing

Buchreihe : Springer INdAM Series

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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This book contains the contributions resulting from the 6th Italian-Japanese workshop on Geometric Properties for Parabolic and Elliptic PDEs, which was held in Cortona (Italy) during the week of May 20–24, 2019. This book will be of great interest for the mathematical community and in particular for researchers studying parabolic and elliptic PDEs. It covers many different fields of current research as follows: convexity of solutions to PDEs, qualitative properties of solutions to parabolic equations, overdetermined problems, inverse problems, Brunn-Minkowski inequalities, Sobolev inequalities, and isoperimetric inequalities.

Inhaltsverzeichnis

Frontmatter

Poincaré and Hardy Inequalities on Homogeneous Trees

Abstract

We study Hardy-type inequalities on infinite homogeneous trees. More precisely, we derive optimal Hardy weights for the combinatorial Laplacian in this setting and we obtain, as a consequence, optimal improvements for the Poincaré inequality.

Elvise Berchio, Federico Santagati, Maria Vallarino

Ground State Solutions for the Nonlinear Choquard Equation with Prescribed Mass

Abstract

We study existence of radially symmetric solutions for the nonlocal problem:

https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-73363-6_2/MediaObjects/508397_1_En_2_Figa_HTML.gif

where N ≥ 3, α ∈ (0, N), c > 0, $I_\alpha (x)={A_\alpha \over |x|{ }^{N-\alpha }}$ is the Riesz potential, $F\in C^1(\mathbb {R},\mathbb {R}), F'(s) = f(s)$, μ is a unknown Lagrange multiplier. Using a Lagrange formulation of the problem (1 ), we develop new deformation arguments under a version of the Palais-Smale condition introduced in the recent papers (Hirata and Tanaka, Adv Nonlinear Stud 19:263–290, 2019; Ikoma and Tanaka, Adv Differ Equ 24:609–646, 2019) and we prove the existence of a ground state solution for the nonlinear Choquard equation with prescribed mass, when F satisfies Berestycki-Lions type conditions.

Silvia Cingolani, Kazunaga Tanaka

Optimization of the Structural Performance of Non-homogeneous Partially Hinged Rectangular Plates

Abstract

We consider a non-homogeneous partially hinged rectangular plate having structural engineering applications. In order to study possible remedies for torsional instability phenomena we consider the gap function as a measure of the torsional performances of the plate. We treat different configurations of load and we study which density function is optimal for our aims. The analysis is in accordance with some results obtained studying the corresponding eigenvalue problem in terms of maximization of the ratio of specific eigenvalues. Some numerical experiments complete the analysis.

Alessio Falocchi

Energy-Like Functional in a Quasilinear Parabolic Chemotaxis System

Abstract

This note deals with a one-dimensional quasilinear chemotaxis system. The first part summarizes recent results, in which a new energy-like functional is introduced and plays a key role. In the latter half, the energy-like functional will be derived in a more general situation.

Kentaro Fujie

Solvability of a Semilinear Heat Equation via a Quasi Scale Invariance

Abstract

Solvability of semilinear heat equations with general nonlinearity is investigated. Applying a quasi scale invariant transformation, we clarify the threshold singularity of initial data for existence and nonexistence results.

Yohei Fujishima, Norisuke Ioku

Bounds for Sobolev Embedding Constants in Non-simply Connected Planar Domains

Abstract

In a bounded non-simply connected planar domain Ω, with a boundary split in an interior part and an exterior part, we obtain bounds for the embedding constants of some subspaces of H ¹( Ω) into L ^p( Ω) for any p > 1, p ≠ 2. The subspaces contain functions which vanish on the interior boundary and are constant (possibly zero) on the exterior boundary. We also evaluate the precision of the obtained bounds in the limit situation where the interior part tends to disappear and we show that it does not depend on p. Moreover, we emphasize the failure of symmetrization techniques in these functional spaces. In simple situations, a new phenomenon appears: the existence of a break even surface separating masses for which symmetrization increases/decreases the Dirichlet norm. The question whether a similar phenomenon occurs in more general situations is left open.

Filippo Gazzola, Gianmarco Sperone

Sharp Estimate of the Life Span of Solutions to the Heat Equation with a Nonlinear Boundary Condition

Abstract

Consider the heat equation with a nonlinear boundary condition

$$\displaystyle \mathrm {(P)}\qquad \left \{ \begin {array}{ll} \partial _t u=\Delta u,\qquad & x\in {\mathbf {R}}^N_+,\,\,\,t>0,\\ \displaystyle {-\frac {\partial u}{\partial x_N} u}=u^p, & x\in \partial {\mathbf {R}}^N_+,\,\,\,t>0,\\ u(x,0)=\kappa \psi (x),\qquad & x\in \overline {{\mathbf {R}}^N_+}, \end {array} \right . \qquad \qquad $$

where N ≥ 1, p > 1, κ > 0 and ψ is a nonnegative measurable function in ${\mathbf {R}}^N_+ :=\{y\in {\mathbf {R}}^N:y_N>0 \}$. Let us denote by T(κψ) the life span of solutions to problem (P). We investigate the relationship between the singularity of ψ at the origin and T(κψ) for sufficiently large κ > 0 and the relationship between the behavior of ψ at the space infinity and T(κψ) for sufficiently small κ > 0. Moreover, we obtain sharp estimates of T(κψ), as κ →∞ or κ → +0.

Kotaro Hisa

Neutral Inclusions, Weakly Neutral Inclusions, and an Over-determined Problem for Confocal Ellipsoids

Abstract

An inclusion is said to be neutral to uniform fields if upon insertion into a homogenous medium with a uniform field it does not perturb the uniform field at all. It is said to be weakly neutral if it perturbs the uniform field mildly. Such inclusions are of interest in relation to invisibility cloaking and effective medium theory. There have been some attempts lately to construct or to show existence of such inclusions in the form of core-shell structure or a single inclusion with the imperfect bonding parameter attached to its boundary. The purpose of this paper is to review recent progress in such attempts. We also discuss about the over-determined problem for confocal ellipsoids which is closely related with the neutral inclusion, and its equivalent formulation in terms of Newtonian potentials. The main body of this paper consists of reviews on known results, but some new results are also included.

Yong-Gwan Ji, Hyeonbae Kang, Xiaofei Li, Shigeru Sakaguchi

Nonexistence of Radial Optimal Functions for the Sobolev Inequality on Cartan-Hadamard Manifolds

Abstract

It is well known that the Euclidean Sobolev inequality holds on any Cartan-Hadamard manifold of dimension n ≥ 3, i.e. any complete, simply connected Riemannian manifold with nonpositive sectional curvature. Moreover, in the very special case of the Euclidean space itself, the optimal constant is achieved by the Aubin-Talenti functions. On a generic Cartan-Hadamard manifold $ \mathbb {M}^n $, one may ask whether there exist at all optimal functions. Here we prove, with ad hoc arguments that do not take advantage of the validity of the so-called Cartan-Hadamard conjecture (claiming that such optimal constant is always Euclidean), that this is false at least for functions that are radially symmetric with respect to the geodesic distance from a fixed pole. More precisely, we show that if the optimum in the Sobolev inequality is achieved by some radial function, then $\mathbb {M}^n $ must be isometric to $ \mathbb {R}^n $.

Tatsuki Kawakami, Matteo Muratori

Semiconvexity of Viscosity Solutions to Fully Nonlinear Evolution Equations via Discrete Games

Abstract

In this paper, by using a discrete game interpretation of fully nonlinear parabolic equations proposed by Kohn and Serfaty (Commun Pure Appl Math 63(10):1298–1350, 2010), we show that the spatial semiconvexity of viscosity solutions is preserved for a class of fully nonlinear evolution equations with concave parabolic operators. We also reduce the game-theoretic argument to the viscous and inviscid Hamilton-Jacobi equations, categorizing the semiconvexity regularity of solutions in terms of semiconcavity of the Hamiltonian.

Qing Liu

An Interpolating Inequality for Solutions of Uniformly Elliptic Equations

Abstract

We extend an inequality for harmonic functions, obtained in Magnanini and Poggesi (Calc Var Partial Differ Equ 59(1):Paper No. 35, 2020) and Poggesi (The Soap Bubble Theorem and Serrin’s problem: quantitative symmetry, PhD thesis, Università di Firenze, 2019), to the case of solutions of uniformly elliptic equations in divergence form, with merely measurable coefficients. The inequality for harmonic functions turned out to be a crucial ingredient in the study of the stability of the radial symmetry for Alexandrov’s Soap Bubble Theorem and Serrin’s problem. The proof of our inequality is based on a mean value property for elliptic operators stated and proved in Caffarelli (The Obstacle Problem. Lezioni Fermiane. [Fermi Lectures]. Accademia Nazionale dei Lincei, Rome; Scuola Normale Superiore, Pisa, 1998) and Blank and Hao (Commun Anal Geom 23(1):129–158, 2015).

Rolando Magnanini, Giorgio Poggesi

Asymptotic Behavior of Solutions for a Fourth Order Parabolic Equation with Gradient Nonlinearity via the Galerkin Method

Abstract

In this paper we consider the initial-boundary value problem for a fourth order parabolic equation with gradient nonlinearity. The problem is regarded as the L ²-gradient flow for an energy functional which is unbounded from below. We first prove the existence and the uniqueness of solutions to the problem via the Galerkin method. Moreover, combining the potential well method with the Galerkin method, we study the asymptotic behavior of global-in-time solutions to the problem.

Nobuhito Miyake, Shinya Okabe

A Note on Radial Solutions to the Critical Lane-Emden Equation with a Variable Coefficient

Abstract

In this note, we consider the following problem

$$\displaystyle \begin {cases} -\Delta u=(1+g(x))u^{\frac {N+2}{N-2}},\ u>0\text{ in }B,\\ u=0\text{ on }\partial B, \end {cases} $$

where N ≥ 3 and $B\subset \mathbb {R}^N$ is the unit ball centered at the origin and g(x) is a radial Hölder continuous function such that g(0) = 0. We prove the existence and nonexistence of radial solutions by the variational method with the concentration compactness analysis and the Pohozaev identity.

Daisuke Naimen, Futoshi Takahashi

Remark on One Dimensional Semilinear Damped Wave Equation in a Critical Weighted L 2-space

Abstract

We study the Cauchy problem of the semilinear damped wave equation in one space dimension. We show the existence of global solutions in the critical case with small initial data in weighted L ²-spaces. This problem in multidimensional cases was dealt with in Sobajima (Differ Integr Equ 32:615–638, 2019) via the weighted Hardy inequality which is false in one-dimension. The crucial idea of the proof is the use of an incomplete version of Hardy inequality.

Motohiro Sobajima, Yuta Wakasugi

Titel: Geometric Properties for Parabolic and Elliptic PDE's
herausgegeben von: Prof. Vincenzo Ferone
Prof. Tatsuki Kawakami
Prof. Paolo Salani
Prof. Futoshi Takahashi
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-73363-6
Print ISBN: 978-3-030-73362-9
DOI: https://doi.org/10.1007/978-3-030-73363-6

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Inhaltsverzeichnis

Frontmatter

Poincaré and Hardy Inequalities on Homogeneous Trees

Ground State Solutions for the Nonlinear Choquard Equation with Prescribed Mass

Optimization of the Structural Performance of Non-homogeneous Partially Hinged Rectangular Plates

Energy-Like Functional in a Quasilinear Parabolic Chemotaxis System

Solvability of a Semilinear Heat Equation via a Quasi Scale Invariance

Bounds for Sobolev Embedding Constants in Non-simply Connected Planar Domains

Sharp Estimate of the Life Span of Solutions to the Heat Equation with a Nonlinear Boundary Condition

Neutral Inclusions, Weakly Neutral Inclusions, and an Over-determined Problem for Confocal Ellipsoids

Nonexistence of Radial Optimal Functions for the Sobolev Inequality on Cartan-Hadamard Manifolds

Semiconvexity of Viscosity Solutions to Fully Nonlinear Evolution Equations via Discrete Games

An Interpolating Inequality for Solutions of Uniformly Elliptic Equations

Asymptotic Behavior of Solutions for a Fourth Order Parabolic Equation with Gradient Nonlinearity via the Galerkin Method

A Note on Radial Solutions to the Critical Lane-Emden Equation with a Variable Coefficient

Remark on One Dimensional Semilinear Damped Wave Equation in a Critical Weighted L 2-space

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