Existence of radial solution for a quasilinear equation with singular nonlinearity

We prove that the equation \begin{eqnarray*} -\Delta_p u =\lambda\Big( \frac{1} {u^\delta} + u^q + f(u)\Big)\;\text{ in } \, B_R(0) u =0 \,\text{ on} \; \partial B_R(0), \quad u>0 \text{ in } \, B_R(0) \end{eqnarray*} admits a weak radially symmetric solution for $\lambda>0$ sufficiently small, $0<\delta<1$ and $p-1<q<p^{*}-1$. We achieve this by combining a blow-up argument and a Liouville type theorem to obtain a priori estimates for the regularized problem. Using a variant of a theorem due to Rabinowitz we derive the solution for the regularized problem and then pass to the limit.Comment: 16 page

Regularity of elastic fields in composites

It is well known that high stress concentrations can occur in elastic composites in particular due to the interaction of geometrical singularities like corners, edges and cracks and structural singularities like jumping material parameters. In the project C5 "Stress concentrations in heterogeneous materials" of the SFB 404 "Multifield Problems in Solid and Fluid Mechanics" it was mathematically analyzed where and which kind of stress singularities in coupled linear and nonlinear elastic structures occur. In the linear case asymptotic expansions near the geometrical and structural peculiarities are derived, formulae for generalized stress intensity factors included. In the nonlinear case such expansions are unknown in general and regularity results are proved for elastic materials with power-law constitutive equations with the help of the difference quotient technique combined with a quasi-monotone covering condition for the subdomains and the energy densities. Furthermore, some applications of the regularity results to shape and structure optimization and the Griffith fracture criterion in linear and nonlinear elastic structures are discussed. Numerical examples illustrate the results

Existence theorems for a crystal surface model involving the p-Laplace operator

The manufacturing of crystal films lies at the heart of modern nanotechnology. How to accurately predict the motion of a crystal surface is of fundamental importance. Many continuum models have been developed for this purpose, including a number of PDE models, which are often obtained as the continuum limit of a family of kinetic Monte Carlo models of crystal surface relaxation that includes both the solid-on-solid and discrete Gaussian models. In this paper we offer an analytical perspective into some of these models. To be specific, we study the existence of a weak solution to the boundary value problem for the equation - \Delta e^{-\mbox{div}\left(|\nabla u|^{p-2}\nabla u\right)}+au=f, where $p>1, a>0$ are given numbers and $f$ is a given function. This problem is derived from a crystal surface model proposed by J.L.~Marzuola and J.~Weare (2013 Physical Review, E 88, 032403). The mathematical challenge is due to the fact that the principal term in our equation is an exponential function of a p-Laplacian. Existence of a suitably-defined weak solution is established under the assumptions that $p\in(1,2], \ N\leq 4$, and $f\in W^{1,p}$. Our investigations reveal that the key to our existence assertion is how to control the set where -\mbox{div}\left(|\nabla u|^{p-2}\nabla u\right) is $\pm\infty$

Quelques contributions au calcul des variations et aux équations elliptiques

Les travaux de recherche qui sont présentés font partie de trois thématiques différentes. Le premier sujet concerne les systèmes d’une classe d'équations aux dérivées partielles, dites «implicites» dans la littérature. Ces problèmes sont complètement non linéaires. Les équations scalaires, où l'inconnue est une fonction, admettent en général, une infinité de solutions. On développe des méthodes variationnelles pour sélectionner des solutions avec des critères de régularités. On traite aussi les cadres vectoriels, où l'inconnue est une application, en présentant des théorèmes d’existence et quelques applications.Les problèmes isopérimétriques font partie de la deuxième thématique de recherche. On traite des inégalités isopérimétriques pour des problèmes aux valeurs propres non linéaires, ainsi que la version quantitative de l'inégalité isopérimétrique classique. On étudie aussi les propriétés de symétrie desminimiseurs d’un problème variationnel non coercitif sur une boule, en montrant une rupture de symétrie, en fonction de l’un des paramètres qui définissent le problème.La mauvaise coercitivité est aussi liée au troisième axe de recherche présenté. On analyse des résultats d’existence et régularité de solutions de certains problèmes elliptiques, définis à travers un opérateur elliptique à coercitivité dégénérée. On montre en particulier les effets régularisants dequelques termes d'ordre inférieur pour les problèmes de Dirichlet correspondants, en fonction de la régularité de la source

Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions

Assuming $B_{R}$ is a ball in $\mathbb R^{N}$, we analyze the positive solutions of the problem $$\begin{cases} -\Delta u+u= |u|^{p-2}u, &\text{ in } B_{R},\newline \partial_{\nu}u=0,&\text{ on } \partial B_{R}, \end{cases}$$ that branch out from the constant solution $u=1$ as $p$ grows from $2$ to $+\infty$. The non-zero constant positive solution is the unique positive solution for $p$ close to $2$. We show that there exist arbitrarily many positive solutions as $p\to\infty$ (in particular, for supercritical exponents) or as $R \to \infty$ for any fixed value of $p>2$, answering partially a conjecture in [Bonheure-Noris-Weth]. We give the explicit lower bounds for $p$ and $R$ so that a given number of solutions exist. The geometrical properties of those solutions are studied and illustrated numerically. Our simulations motivate additional conjectures. The structure of the least energy solutions (among all or only among radial solutions) and other related problems are also discussed.Comment: 37 pages, 24 figure

Space dependent adhesion forces mediated by transient elastic linkages : new convergence and global existence results

In the first part of this work we show the convergence with respect to an asymptotic parameter {\epsilon} of a delayed heat equation. It represents a mathematical extension of works considered previously by the authors [Milisic et al. 2011, Milisic et al. 2016]. Namely, this is the first result involving delay operators approximating protein linkages coupled with a spatial elliptic second order operator. For the sake of simplicity we choose the Laplace operator, although more general results could be derived. The main arguments are (i) new energy estimates and (ii) a stability result extended from the previous work to this more involved context. They allow to prove convergence of the delay operator to a friction term together with the Laplace operator in the same asymptotic regime considered without the space dependence in [Milisic et al, 2011]. In a second part we extend fixed-point results for the fully non-linear model introduced in [Milisic et al, 2016] and prove global existence in time. This shows that the blow-up scenario observed previously does not occur. Since the latter result was interpreted as a rupture of adhesion forces, we discuss the possibility of bond breaking both from the analytic and numerical point of view