740 research outputs found
Partial Regularity for Stationary Solutions to Liouville-Type Equation in dimension 3
In dimension , we prove that the singular set of any stationary solution
to the Liouville equation , which belongs to , 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
or on the sphere with values into a closed manifold .
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 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 with respect
to the trace of the M\"obius transformations of the dimensional disk we
derive a countable family of relations involving the Fourier coefficients of
weak half-harmonic maps In the same spirit we also
provide as many Pohozaev-type identities in -D for stationary harmonic maps
as conformal vector fields in 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 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 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
at infinity. This latter case encompasses some equations related
to backward stochastic differential equations
Blow-up analysis of a nonlocal Liouville-type equation
In this paper we perform a blow-up and quantization analysis of the following
nonlocal Liouville-type equation \begin{equation}(-\Delta)^\frac12 u= \kappa
e^u-1~\mbox{in ,} \end{equation} where stands for
the fractional Laplacian and is a bounded function. We interpret the
above equation as the prescribed curvature equation to a curve in conformal
parametrization. We also establish a relation between this equation and the
analogous equation in \begin{equation}
(-\Delta)^\frac{1}{2} u =Ke^u \quad \text{in }\mathbb{R}, \end{equation} with
bounded on .Comment: 59 page
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]
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