1,720,972 research outputs found
Some characterizations of magnetic Sobolev spaces
Full text linkThe aim of this note is to survey recent results contained in Nguyen H-M, Squassina M. [On anisotropic Sobolev spaces. Commun Contemp Math, to appear. DOI:10.1142/S0219199718500177]; Nguyen H-M, Pinamonti A, Squassina M, et al. [New characterizations of magnetic Sobolev spaces. Adv Nonlinear Anal. 2018;7(2):227–245]; Pinamonti A, Squassina M, Vecchi E. [Magnetic BV functions and the Bourgain-Brezis-Mironescu formula. Adv Calc Var, to appear. DOI:10.1515/acv-2017-0019]; Pinamonti A, Squassina M, Vecchi E. [The Maz'ya-Shaposhnikova limit in the magnetic setting. J Math Anal Appl. 2017;449:1152–1159] and Squassina M, Volzone B. [Bourgain-Brezis-Mironescu formula for magnetic operators. C R Math Acad Sci Paris. 2016;354:825–831], where the authors extended to the magnetic setting several characterizations of Sobolev and BV functions
Geometric aspects of p-capacitary potentials
No full textWe provide monotonicity formulas for solutions to the p-Laplace equation defined in the exterior of a convex domain. A number of analytic and geometric consequences are derived, including the classical Minkowski inequality as well as new characterizations of rotationally symmetric solutions and domains. The proofs rely on the conformal splitting technique introduced by the second author in collaboration with V. Agostiniani
Strict starshapedness of solutions to the horizontal p-Laplacian in the Heisenberg group
Full text linkWe examine the geometry of the level sets of particular horizontally p-harmonic functions in the Heisenberg group. We find sharp, natural geometric conditions ensuring that the level sets of the p-capacitary potential of a bounded annulus in the Heisenberg group are strictly starshaped
New integral estimates in substatic Riemannian manifolds and the Alexandrov Theorem
No full textWe derive new integral estimates on substatic manifolds with boundary of horizon type, naturally arising in General Relativity. In particular, we generalize to this setting an identity due to Magnanini-Poggesi [24] leading to the Alexandrov Theorem in Rn and improve on a Heintze-Karcher type inequality due to Li-Xia [22]. Our method relies on the introduction of a new vector field with nonnegative divergence, generalizing to this setting the P-function technique of Weinberger [36]
Towards a Brezis–Oswald-type result for fractional problems with Robin boundary conditions
Full text linkWe consider a boundary value problem driven by the p-fractional Laplacian with nonlocal Robin boundary conditions and we provide necessary and sufficient conditions which ensure the existence of a unique positive (weak) solution. The results proved in this paper can be considered a first step towards a complete generalization of the classical result by Brezis and Oswald (Nonlinear Anal 10:55–64, 1986) to the nonlocal setting
Pohozaev-type identities for differential operators driven by homogeneous vector fields
Full text linkWe prove Pohozaev-type identities for smooth solutions of Euler-Lagrange equations of second and fourth order that arise from functional a depending on homogeneous Hörmander vector fields. We then exploit such integral identities to prove non-existence results for the associated boundary value problems
Sublinear Equations Driven by Hörmander Operators
Full text linkWe characterize the existence of a unique positive weak solution for a Dirichlet boundary value problem driven by a linear second-order differential operator modeled on Hörmander vector fields, where the right hand side has sublinear growth
The Equality Case in the Substatic Heintze–Karcher Inequality
Full text linkWe provide a rigidity statement for the equality case of the Heintze–Karcher inequality in substatic manifolds. We apply such a result in the warped product setting to fully remove assumption (H4) in the celebrated Brendle’s characterization of constant mean curvature hypersurfaces in warped products
Existence and uniqueness theorems for some semi-linear equations on locally finite graphs
Full text linkWe study some semi-linear equations for the -Laplacian operator on
locally finite weighted graphs. We prove existence of weak solutions for all
and via a variational method already known
in the literature by exploiting the continuity properties of the energy
functionals involved. When , we also establish a uniqueness result in the
spirit of the Brezis-Strauss Theorem. We finally provide some applications of
our main results by dealing with some Yamabe-type and Kazdan-Warner-type
equations on locally finite weighted graphs.Comment: 13 page
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