We consider nonlocal linear Schrödinger-type critical systems of the type(1)where Ω is antisymmetric potential in L 2 (IR, so(m)), v is a IR m valued map and Ω v denotes the matrix multiplication. We show that every solution v ∈ L 2 (IR, IR m ) of (1) is in fact in L p loc (IR, IR m ), for every 2 ≤ p < +∞, in other words, we prove that the system (1) which is a-priori only critical in L 2 happens to have a subcritical behavior for antisymmetric potentials. As an application we obtain the C 0,α loc regularity of weak 1/2-harmonic maps into C 2 compact manifold without boundary.

We prove a Strong Maximum Principle for semicontinuous viscosity subsolutions or supersolutions of fully nonlinear degenerate elliptic PDEs, which complements the results of [17]. Our assumptions and conclusions are different from those in [17], in particular our maximum principle implies the nonexistence of a dead core. We test the assumptions on several examples involving the p-Laplacian and the minimal surface operator, and they turn out to be sharp in all cases where the existence of a dead core is known. We can also cover equations that are singular for p 0 and very degenerate operators such as the I -Laplacian and some first order Hamilton-Jacobi operators.

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In this paper, we prove a comparison result between semicontinuous viscosity sub and supersolutions growing at most quadratically of second-order degenerate parabolic Hamilton-Jacobi-Bellman and Isaacs equations. As an application, we characterize the value function of a finite horizon stochastic control problem with unbounded controls as the unique viscosity solution of the corresponding dynamic programming equation.

Abstract. We prove the convergence at a large scale of a non-local first order equation to an anisotropic mean curvature motion. The equation is an eikonal-type equation with a velocity depending in a non-local way on the solution itself, which arises in the theory of dislocation dynamics. We show that if an anisotropic mean curvature motion is approximated by equations of this type then it is always of variational type, whereas the converse is true only in dimension two.

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 $S^1$,} \end{equation} where $(-\Delta)^\frac{1}{2}$ stands for the fractional Laplacian and $\kappa$ 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 $\mathbb{R}$ \begin{equation} (-\Delta)^\frac{1}{2} u =Ke^u \quad \text{in }\mathbb{R}, \end{equation} with $K$ bounded on $\mathbb{R}$.Comment: 59 page

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

Abstract. We study uniformly elliptic fully nonlinear equationsand prove results of Gidas-Ni-Nirenberg type for positive viscosity solutions of such equations. We show that symmetries of the equation and the domain are reflected by the solution, both in bounded and unbounded domains.