Quaas, Alexander

We study existence of principal eigenvalues of a fully nonlinear integro-differential elliptic equations with a drift term via the Krein–Rutman theorem and regularity estimates up to boundary of viscosity solutions. We also show simplicity of eigenfunctions in the viscosity sense by using a nonlocal version of the ABP estimate and a “sweeping lemma”.

In this paper we study the evolution problem associated with the first fractional eigenvalue. We prove that the Dirichlet problem with homogeneous boundary condition is well posed for this operator in the framework of viscosity solutions (the problem has existence and uniqueness of a solution and a comparison principle holds). In addition, we show that solutions decay to zero exponentially fast as t → ∞ with a bound that is given by the first eigenvalue for this problem that we also study

We study whether the solutions of a fully nonlinear, uniformly parabolic equation with superquadratic growth in the gradient satisfy initial and homogeneous boundary conditions in the classical sense, a problem we refer to as the classical Dirichlet problem. Our main results are: the nonexistence of global-in-time solutions of this problem, depending on a specific largeness condition on the initial data, and the existence of local-in-time solutions for initial data C1 up to the boundary. Global existence is know when boundary conditions are understood in the viscosity sense, what is known as the generalized Dirichlet problem. Therefore, our result implies loss of boundary conditions in finite time. Specifically, a solution satisfying homogeneous boundary conditions in the viscosity sense eventually becomes strictly positive at some point of the boundary.

In this paper, we study the existence and uniqueness of positive solutions for the following nonlinear fractional elliptic equation: (−∆)αu = λa(x)u − b(x)u p in R N , where α ∈ (0, 1), N ≥ 2, λ > 0, a and b are positive smooth function in R N satisfying a(x) → a ∞ > 0 and b(x) → b ∞ > 0 as |x| → ∞. Our proof is based on a comparison principle and existence, uniqueness and asymptotic behaviors of various boundary blow-up solutions for a class of elliptic equations involving the fractional Laplacian.