1,721,001 research outputs found
Schauder estimates for Kolmogorov-Fokker-Planck operators with coefficients measurable in time and Hölder continuous in space
Full text linkWe consider degenerate Kolmogorov-Fokker-Planck operators Lu=∑ I, j = 1 q a_ij (x,t) ∂ x_i x_j ^ 2 u+∑ k, j = 1 N b_jk x_k ∂ x_j u − ∂_t u,(x,t)∈R^N+1, N ≥ q ≥ 1 such that the corresponding model operator having constant a_ij is hypoelliptic, translation invariant w.r.t. a Lie group operation in R^N+1 and 2-homogeneous w.r.t. a family of nonisotropic dilations. The coefficients a_ij are bounded and Hölder continuous in space (w.r.t. some distance induced by L in R^N) and only bounded measurable in time; the matrix {a_ij} I, j = 1 q is symmetric and uniformly positive on R^q. We prove “partial Schauder a priori estimates” of the kind ∑ I, j = 1 q ‖∂ x_i x_j ^ 2 u‖C_x^α(S_T)+‖Y_u‖C_x^α(S_T) ≤ c {‖Lu‖C_x^α(S_T) + ‖u‖C^0(S_T)} for suitable functions u, where Yu=∑ k, j = 1 N b_jk x_k ∂ x_j u−∂ t u and ‖f‖C_x^α(S_T) = sup(t ≤ T_x1,x2∈R^N), sup(x1≠x2) |f(x_1, t), f(x_2, t)|/||x_1-x_2||^α +‖f‖L^∞(S_T). We also prove that the derivatives ∂ x_i x_j^2 u are locally Hölder continuous in space and time while ∂ x_i u and u are globally Hölder continuous in space and time
A Liouville theorem for elliptic equations with a potential on infinite graphs
Full text linkBiagi S, Meglioli G, Punzo F. A Liouville theorem for elliptic equations with a potential on infinite graphs. Calculus of Variations and Partial Differential Equations . 2024;63(7): 165.We investigate the validity of the Liouville property for a class of elliptic equations with a potential, posed on infinite graphs. Under suitable assumptions on the graph and on the potential, we prove that the unique bounded solution is u equivalent to 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}. We also show that on a special class of graphs the condition on the potential is optimal, in the sense that if it fails, then there exist infinitely many bounded solutions
Left-Invariance for Smooth Vector Fields and Applications
Full text linkLet X = {X_0,..., X_m} be a family of smooth vector fields on an open set A ⊆ R^N. Motivated by applications to the PDE theory of Hörmander operators, for a suitable class of open sets A, we find necessary and sufficient conditions on X for the existence of a Lie group (A, ∗) such that the operator L = X_1^2 + ... + X_m^2 + X_0 is left-invariant with
respect to the operation ∗. Our approach is constructive, as the group law is constructed
by means of the solution of a suitable ODE naturally associated to vector fields in X.
We provide an application to a partial differential operator appearing in the Finance
Global heat kernels for parabolic homogeneous hörmander operators
No full textThe aim of this paper is to prove the existence and several selected properties of a global fundamental Heat kernel Gamma for the parabolic operators H = Sigma(m)(j=1) X-j(2)-partial derivative(t), where X-1,..., X-m are smooth vector fields on R-n satisfying Hormander's rank condition, and enjoying a suitable homogeneity assumption with respect to a family of non-isotropic dilations. The proof of the existence of G is based on a (algebraic) global lifting technique, together with a representation of G in terms of the integral (performed over the lifting variables) of the Heat kernel for the Heat operator associated with a suitable sub-Laplacian on a homogeneous Carnot group. Among the features of G we prove: homogeneity and symmetry properties; summability properties; its vanishing at infinity; the uniqueness of the bounded solutions of the related Cauchy problem; reproduction and density properties; an integral representation for the higher- order derivatives
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
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
Boundary value problems associated with singular strongly nonlinear equations with functional terms
Full text linkWe study boundary value problems associated with singular, strongly nonlinear differential equations with functional terms of type (φ (k (t) x ′ (t))) ′ + f (t, G x (t)) ρ (t, x ′ (t)) = 0, on a compact interval [a, b]. These equations are quite general due to the presence of a strictly increasing homeomorphism φ, the so-called φ-Laplace operator, of a non-negative function k, which may vanish on a set of null measure, and moreover of a functional term Gx. We look for solutions, in a suitable weak sense, which belong to the Sobolev space W1,1([a, b]). Under the assumptions of the existence of a well-ordered pair of upper and lower solutions and of a suitable Nagumo-type growth condition, we prove an existence result by means of fixed point arguments
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)
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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