Partial Regularity for Stationary Solutions to Liouville-Type Equation in dimension 3

In dimension $n=3$, we prove that the singular set of any stationary solution to the Liouville equation $-\Delta u=e^u$, which belongs to $W^{1,2}$, has Hausdorff dimension at most 1.Comment: 20 page

Some Remarks on Pohozaev-Type Identities

The aim of this note is to discuss in more detail the Pohozaev-type identities that have been recently obtained by the author, Paul Laurain and Tristan Rivi\`ere in the framework of half-harmonic maps defined either on $R$ or on the sphere $S^1$ with values into a closed manifold $N^n\subset R^m$. Weak half-harmonic maps are critical points of the following nonlocal energy \int_{R}|(-\Delta)^{1/4}u|^2 dx~~\mbox{or}~~\int_{S^1}|(-\Delta)^{1/4}u|^2\ d\theta. If $u$ is a sufficiently smooth critical point of the above energy then it satisfies the following equation of stationarity \frac{du}{dx}\cdot (-\Delta)^{1/2} u=0~~\mbox{a.e in $R$}~~\mbox{or}~~\frac{\partial u}{\partial \theta}\cdot (-\Delta)^{1/2} u=0~~\mbox{a.e in $S^1$.} By using the invariance of the equation of stationarity in $S^1$ with respect to the trace of the M\"obius transformations of the $2$ dimensional disk we derive a countable family of relations involving the Fourier coefficients of weak half-harmonic maps $u\colon S^1\to N^n.$ In the same spirit we also provide as many Pohozaev-type identities in $2$-D for stationary harmonic maps as conformal vector fields in $R^2$ generated by holomorphic functions

Remarks on Neumann boundary problems involving Jacobians

In this short note we explore the validity of Wente-type estimates for Neumann boundary problems involving Jacobians. We show in particular that such estimates do not in general hold under the same hypotheses on the data for Dirichlet boundary problems

Uniqueness results for convex Hamilton-Jacobi equations under p>1p>1 growth conditions on data

Unbounded stochastic control problems may lead to Hamilton-Jacobi-Bellman equations whose Hamiltonians are not always defined, especially when the diffusion term is unbounded with respect to the control. We obtain existence and uniqueness of viscosity solutions growing at most like $o(1+|x|^p)$ at infinity for such HJB equations and more generally for degenerate parabolic equations with a superlinear convex gradient nonlinearity. If the corresponding control problem has a bounded diffusion with respect to the control, then our results apply to a larger class of solutions, namely those growing like $O(1+|x|^p)$ at infinity. This latter case encompasses some equations related to backward stochastic differential equations

Large time behavior of solutions to parabolic equations with Neumann boundary conditions

AbstractIn this paper we are interested in the large time behavior as t→+∞ of the viscosity solutions of parabolic equations with nonlinear Neumann type boundary conditions in connection with ergodic boundary problems which have been recently studied by Barles and the author in [G. Barles, F. Da Lio, On the boundary ergodic problem for fully nonlinear equations in bounded domains with general nonlinear Neumann boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linèaire 22 (5) (2005) 521–541]