On the spectrum of the -biharmonic operator.
We discuss the uniqueness of solutions to problems like⎧ λ |u|s-1u - div (|∇u|p-2∇u) = μ on Ω,⎨⎩ u = 0 in ∂Ω,where λ ≥ 0 and μ is a signed Radon measure.
We study the Ambrosetti–Prodi and Ambrosetti–Rabinowitz problems.We prove for the first one the existence of a continuum of solutions with shape of a reflected (-shape). Next, we show that there is a relationship between these two problems.
We study very weak solutions of an A-harmonic equation to show that they are in fact the usual solutions.
In this paper we consider two-dimensional quasilinear equations of the form and study the properties of the solutions u with bounded and non-vanishing gradient. Under a weak assumption involving the growth of the argument of (notice that is a well-defined real function since on ) we prove that is one-dimensional, i.e., for some unit vector . As a consequence of our result we obtain that any solution having one positive derivative is one-dimensional. This result provides a proof of...
We consider on a two-dimensional flat torus defined by a rectangular periodic cell the following equationIt is well-known that the associated energy functional admits a minimizer for each . The present paper shows that these minimizers depend actually only on one variable. As a consequence, setting to be the first eigenvalue of the Laplacian on the torus, the minimizers are identically zero whenever . Our results hold more generally for solutions that are Steiner symmetric, up to a translation....
This paper is devoted to the numerical solution of nonlinear elliptic partial differential equations. Such problems describe various phenomena in science. An approach that exploits Hilbert space theory in the numerical study of elliptic PDEs is the idea of preconditioning operators. In this survey paper we briefly summarize the main lines of this theory with various applications.
In this paper we find the optimal regularity for viscosity solutions of the pseudo infinity Laplacian. We prove that the solutions are locally Lipschitz and show an example that proves that this result is optimal. We also show existence and uniqueness for the Dirichlet problem.