1,720,995 research outputs found
Steiner Formula and Gaussian Curvature in the Heisenberg Group
Full text linkThe classical Steiner formula expresses the volume of the ∈-neighborhood Ω∈ of a bounded and regular domain Ω⊂Rn as a polynomial of degree n in ∈. In particular, the coefficients of this polynomial are the integrals of functions of the curvatures of the boundary ∂Ω. The aim of this note is to present the Heisenberg counterpart of this result. The original motivation for studying this kind of extension is to try to identify a suitable candidate for the notion of horizontal Gaussian curvature. The results presented in this note are contained in the paper [4] written in collaboration with Zoltàn Balogh, Fausto Ferrari, Bruno Franchi and Kevin Wildric
On the mixed local-nonlocal Hénon equation
Full text linkIn this paper we consider a Hénon-type equation driven by a nonlinear operator obtained as a combination of a local and nonlocal term. We prove existence and non-existence akin to the classical result by Ni, and a stability result as the fractional parameter s→1.Fil: Salort, Ariel Martin. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Calculo. - Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Calculo; ArgentinaFil: Vecchi, Eugenio. Politecnico di Milano; Itali
A symmetry result for elliptic systems in punctured domains
Full text linkWeconsideranellipticsystemofequationsinapuncturedbounded domain. We prove that if the domain is convex in one direction and symmetric with respect to the reflections induced by the normal hyperplane to such a direction, then the solution is necessarily symmetric under this reflection and monotone in the corresponding direction. As a consequence, we prove symme- try results also for a related polyharmonic problem of any order with Navier boundary conditions
Intrinsic curvature of curves and surfaces and a Gauss-Bonnet theorem in the Heisenberg group
Full text linkWe use a Riemannnian approximation scheme to define a notion of intrinsic Gaussian curvature for a Euclidean C2 -smooth surface in the Heisenberg group H away from characteristic points, and a notion of intrinsic signed geodesic curvature for Euclidean C2 -smooth curves on surfaces. These results are then used to prove a Heisenberg version of the Gauss–Bonnet theorem. An application to Steiner’s formula for the Carnot–Carathéodory distance in H is provided
Semilinear elliptic equations involving mixed local and nonlocal operators
Full text linkIn this paper, we consider an elliptic operator obtained as the superposition of a classical second-order differential operator and a nonlocal operator of fractional type. Though the methods that we develop are quite general, for concreteness we focus on the case in which the operator takes the form − Δ + ( − Δ)s, with s ∈ (0, 1). We focus here on symmetry properties of the solutions and we prove a radial symmetry result, based on the moving plane method, and a one-dimensional symmetry result, related to a classical conjecture by G.W. Gibbons
ON A BREZIS-OSWALD-TYPE RESULT FOR DEGENERATE KIRCHHOFF PROBLEMS
Full text linkIn the present note we establish an almost-optimal solvability result for Kirchhoff-type problems of the following form{--M (||Delta u||L2(Omega)) = u= fz(x,u) in f(x, u) in Omega,u >=, L2(omega) u > 0, in Omega u = 0 on partial derivative Omega.partial differential n. where f has sublinear growth and M is a non-decreasing map with M(0) >= 0. Our approach is purely variational, and the result we obtain is resemblant to the one established by Brezis and Oswald (Nonlinear Anal., 1986) for sublinear elliptic equations
Brezis–Nirenberg type results for the anisotropic p‐Laplacian
No full textIn this paper, we consider a quasilinear elliptic and critical problem with Dirichlet boundary conditions in presence of the anisotropic (Formula presented.) -Laplacian. The critical exponent is the usual (Formula presented.) such that the embedding (Formula presented.) is not compact. We prove the existence of a weak positive solution in presence of both a (Formula presented.) -linear and a (Formula presented.) -superlinear perturbation. In doing this, we have to perform several precise estimates of the anisotropic Aubin–Talenti functions which can be of interest for further problems. The results we prove are a natural generalization to the anisotropic setting of the classical ones by Brezis–Nirenberg (Comm. Pure Appl. Math. 36 (1983), 437–477)
Hölder behavior of viscosity solutions of some fully nonlinear equations in the Heisenberg group
Full text linkIn this paper we prove the regularity of bounded and uniformly continuous viscosity solutions of some degenerate fully nonlinear equations in the first Heisenberg group
On mixed local–nonlocal problems with Hardy potential
No full textIn this article, we study the effect of the Hardy potential on existence, uniqueness, and optimal summability of solutions of the mixed local-nonlocal elliptic problem where ω is a bounded domain in containing the origin and δ3 > 0. In particular, we will discuss the existence, non-existence, and uniqueness of solutions in terms of the summability of f and of the value of the parameter δ3
Symmetry and monotonicity of singular solutions to p-Laplacian systems involving a first order term
No full textWe consider positive singular solutions (i.e. with a non-removable singularity) of a system of PDEs driven by p-Laplacian operators and with the additional presence of a nonlinear first order term. By a careful use of a rather new version of the moving plane method, we prove the symmetry of the solutions. The result is already new in the scalar case
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