Some results of nontrivial solutions for a nonlinear PDE in Sobolev space

In this study, we investigate the question of nonexistence of nontrivial solutions of the Robin problem \begin{equation} \left\vert\begin{array}{l} -\dfrac{\partial ^{2}u}{\partial x^{2}}-\sum\limits_{s=1}^{n}\dfrac{\partial}{\partial y_{s}}a_{s}(y,\frac{\partial u}{\partial y_{s}})+f(y,u)=0\text{in }\Omega =\mathbb{R}\times D, \\ \\ u+\varepsilon \dfrac{\partial u}{\partial n}=0\text{ on }\mathbb{R}\times \partial D. \end{array}\right. \tag*{$\left( P\right)$} \end{equation} where $a_{s}:D\times \mathbb{R}\rightarrow \mathbb{R}$ are $H^{1}$-functions with constant sign such that \begin{equation}\begin{array}{c} 2\int\limits_{0}^{\xi _{s}}a_{s}(y,t_{s})dt_{s}-\xi _{s}a_{s}(y,\xi_{s})\leq 0,s=1,...,n \end{array}\tag*{$\left( H_{1}\right)$}\end{equation} and $f:D\times \mathbb{R}\rightarrow \mathbb{R}$ is a real continuous locally Liptschitz function such that \begin{equation} 2F(y,u)-uf(y,u)\leq 0. \tag*{$\left( H_{2}\right)$} \end{equation} We show that the function \begin{equation*} E(x)=\int\limits_{D}\left\vert u(x,y)\right\vert ^{2}dy \end{equation*} is convex on $\mathbb{R}$ . Our proof is based on energy (integral) identities

Spectral Optimization Problems

In this survey paper we present a class of shape optimization problems where the cost function involves the solution of a PDE of elliptic type in the unknown domain. In particular, we consider cost functions which depend on the spectrum of an elliptic operator and we focus on the existence of an optimal domain. The known results are presented as well as a list of still open problems. Related fields as optimal partition problems, evolution flows, Cheeger-type problems, are also considered.Comment: 42 pages with 8 figure

A necessary and sufficient condition for uniqueness of the trivial solution in semilinear parabolic equations

© 2017 Elsevier Inc. In their (1968) paper Fujita and Watanabe considered the issue of uniqueness of the trivial solution of semilinear parabolic equations with respect to the class of bounded, non-negative solutions. In particular they showed that if the underlying ODE has non-unique solutions (as characterised via an Osgood-type condition) and the nonlinearity f satisfies a concavity condition, then the parabolic PDE also inherits the non-uniqueness property. This concavity assumption has remained in place either implicitly or explicitly in all subsequent work in the literature relating to this and other, similar, non-uniqueness phenomena in parabolic equations. In this paper we provide an elementary proof of non-uniqueness for the PDE without any such concavity assumption on f. An important consequence of our result is that uniqueness of the trivial solution of the PDE is equivalent to uniqueness of the trivial solution of the corresponding ODE, which in turn is known to be equivalent to an Osgood-type integral condition on f