arXiv subjects
Publications and source records attributed to Frederic Robert.
We analyze the possible concentration behavior of heat flows related to the Moser-Trudinger energy and derive quantization results completely analogous to the quantization results for solutions of the corresponding elliptic equation. As an application of our results we obtain the existence of critical points of the Moser-Trudinger energy in a supercritical regime.
We consider asymptotic behavior of the following fourth order equation \[ Δ^2 u= ρ\frac{e^{u}}{\int_\Om e^{u} dx} {in} \Om, u= \partial_νu=0 {on} \partial Ω\] where $\Om$ is a smooth oriented bounded domain in $\R^4$. Assuming that $0<ρ\leq C$, we completely characterize the asymptotic behavior of the unbounded solutions.
We investigate the asymptotic behavior as $k \to +\infty$ of sequences $(u_k)_{k\in\mathbb{N}}\in C^4(Ω)$ of solutions of the equations $Δ^2 u_k=V_k e^{4u_k}$ on $Ω$, where $Ω$ is a bounded domain of $\mathbb{R}^4$ and $\lim_{k\to +\infty}V_k=1$ in $C^0_{loc}(Ω)$. The corresponding 2-dimensional problem was studied by Brézis-Merle and Li-Shafrir who pointed out that there is a quantization of the energy when blow-up occurs. As shown by Adimurthi, Struwe and the author, such a quantization does not hold in dimension four for the problem in its full generality. We prove here that under natural hypothesis on $Δu_k$, we recover such a quantization as in dimension 2.
We let $Ω$ be a smooth bounded domain of $\mathbb{R}^4$ and a sequence of fonctions $(V_k)_{k\in\mathbb{N}}\in C^0(Ω)$ such that $\lim_{k\to +\infty}V_k=1$ in $C^0_{loc}(Ω)$. We consider a sequence of functions $(u_k)_{k\in\mathbb{N}}\in C^4(Ω)$ such that $$Δ^2 u_k=V_k e^{4u_k}$$ in $Ω$ for all $k\in\mathbb{N}$. We address in this paper the question of the asymptotic behaviour of the $(u_k)'s$ when $k\to +\infty$. The corresponding problem in dimension 2 was considered by Brézis-Merle and Li-Shafrir (among others), where a blow-up phenomenon was described and where a quantization of this blow-up was proved. Surprisingly, as shown by Adimurthi, Struwe and the author, a similar quantization phenomenon does not hold for this fourth order problem. Assuming that the $u_k$'s are radially symmetrical, we push further the previous analysis. We prove that there are exactly three types of blow-up and we describe each type in a very detailed way.
We solve variationally certain equations of stellar dynamics of the form $-\sum_i\partial_{ii} u(x) =\frac{|u|^{p-2}u(x)}{{\rm dist} (x,{\mathcal A} )^s}$ in a domain $Ω$ of $\rn$, where ${\mathcal A} $ is a proper linear subspace of $\rn$. Existence problems are related to the question of attainability of the best constant in the following recent inequality of Badiale-Tarantello [1]: $$0<μ_{s,¶}(Ω)=\inf{\int_Ω|\nabla u|^2 dx; u\in \huno \hbox{and}\int_Ω\frac{|u(x)|^{\crit(s)}}{|π(x)|^s} dx=1}$$ where $0<s<2$, $\crit(s)=\frac{2(n-s)}{n-2}$ and where $π$ is the orthogonal projection on a linear space $¶$, where $\hbox{dim}_{\rr}¶\geq 2$. We investigate this question and how it depends on the relative position of the subspace $\Porth$, the orthogonal of $¶$, with respect to the domain $Ω$ as well as on the curvature of the boundary $\partialΩ$ at its points of intersection with $\Porth $.
We establish -among other things- existence and multiplicity of solutions for the Dirichlet problem $\sum_i\partial_{ii}u+\frac{|u|^{\crit-2}u}{|x|^s}=0$ on smooth bounded domains $Ω$ of $ \rn$ ($n\geq 3$) involving the critical Hardy-Sobolev exponent $\crit =\frac{2(n-s)}{n-2}$ where $0<s<2$, and in the case where zero (the point of singularity) is on the boundary $\partial Ω$. Just as in the Yamabe-type non-singular framework (i.e., when s=0), there is no nontrivial solution under global convexity assumption (e.g., when $Ω$ is star-shaped around 0). However, in contrast to the non-satisfactory situation of the non-singular case, we show the existence of an infinite number of solutions under an assumption of local strict concavity of $\partial Ω$ at 0 in at least one direction. More precisely, we need the principal curvatures of $\partial Ω$ at 0 to be non-positive but not all vanishing. We also show that the best constant in the Hardy-Sobolev inequality is attained as long as the mean curvature of $\partial Ω$ at 0 is negative, extending the results of [21] and completing our result of [22] to include dimension 3. The key ingredients in our proof are refined concentration estimates which yield compactness for certain Palais-Smale sequences which do not hold in the non-singular case.
In this paper we study the energy function associated to fourth order equations of critical growth on smooth compact conformally flat manifolds of dimension greater or equal than 5.