Elliptic Operators Research Articles

Global Kato smoothing and Strichartz estimates for Schrödinger type equations with rough decay potentials

Let [Formula: see text] be a higher-order elliptic operator on [Formula: see text], where [Formula: see text] is a general bounded decaying potential. This paper focuses on the global Kato smoothing and Strichartz estimates for solutions to Schrödinger-type equation associated with [Formula: see text]. In particular, we first establish sharp global Kato smoothing estimates for [Formula: see text], based on uniform resolvent estimates of Kato–Yajima type for the absolutely continuous part of [Formula: see text]. As a consequence, we also obtain optimal local decay estimates. Using these local decay estimates, we then prove the full set of Strichartz estimates, including the endpoint case. Notably, we derive Strichartz estimates with sharp smoothing effects for higher-order cases with rough potentials, which are applicable to the study of nonlinear higher-order Schrödinger equations. Finally, we introduce new uniform Sobolev estimates of the Kenig–Ruiz–Sogge type, incorporating an additional derivative term, which are crucial for establishing the sharp Kato smoothing estimates.

Notes on peculiarities of the Schwinger-DeWitt technique: One-loop double poles, total-derivative terms, and determinant anomalies

We discuss peculiarities of the Schwinger-DeWitt technique for quantum effective action, associated with the origin of dimensionally regularized double-pole divergences of the one-loop functional determinant for massive Proca model in a curved spacetime. These divergences have the form of the total-derivative term generated by integration by parts in the functional trace of the heat kernel for the Proca vector field operator. Because of the nonminimal structure of second-order derivatives in this operator, its vector field heat kernel has a nontrivial form, involving the convolution of the scalar d’Alembertian Green’s function with its heat kernel. Moreover, its asymptotic expansion is very different from the universal predictions of Gilkey-Seeley heat kernel theory because the Proca operator violates one of the basic assumptions of this theory—the nondegeneracy of the principal symbol of an elliptic operator. This modification of the asymptotic expansion explains the origin of double-pole total-derivative terms. Another hypostasis of such terms is in the problem of multiplicative determinant anomalies—lack of factorization of the functional determinant of a product of differential operators into the product of their individual determinants. We demonstrate that this anomaly should have the form of total-derivative terms and check this statement by calculating divergent parts of functional determinants for products of minimal and nonminimal second-order differential operators in curved spacetime. Published by the American Physical Society 2024

Boundary Hölder continuity of stable solutions to semilinear elliptic problems in 𝐶1,1 domains

Abstract This article establishes the boundary Hölder continuity of stable solutions to semilinear elliptic problems in the optimal range of dimensions n ≤ 9 n\leq 9 , for C 1 , 1 C^{1\smash{,}1} domains. We consider equations − L ⁢ u = f ⁢ ( u ) -Lu=f(u) in a bounded C 1 , 1 C^{1\smash{,}1} domain Ω ⊂ R n \Omega\subset\mathbb{R}^{n} , with u = 0 u=0 on ∂ Ω \partial\Omega , where 𝐿 is a linear elliptic operator with variable coefficients and f ∈ C 1 f\in C^{1} is nonnegative, nondecreasing, and convex. The stability of 𝑢 amounts to the nonnegativity of the principal eigenvalue of the linearized equation − L − f ′ ⁢ ( u ) -L-f^{\prime}(u) . Our result is new even for the Laplacian, for which [X. Cabré, A. Figalli, X. Ros-Oton and J. Serra, Stable solutions to semilinear elliptic equations are smooth up to dimension 9, Acta Math. 224 (2020), 2, 187–252] proved the Hölder continuity in C 3 C^{3} domains.

Monotonicity, asymptotics and level sets for principal eigenvalues of some elliptic operators with shear flow

We investigate the joint effects of diffusion and advection on principal eigenvalues of some elliptic operators with shear flow. Some monotonicity and asymptotic behaviors of principal eigenvalues, with respect to diffusion rate and flow amplitude, are established. These analyses lead to a classification of topological structures of level sets for principal eigenvalues, as a function of diffusion rate and flow amplitude. Our analytical results provide a unifying viewpoint to understand mixing enhancement and dispersal-induced growth, which are apparently two unrelated phenomena, one in fluid mechanics and the other in population dynamics. In our analysis, some limiting Hamilton-Jacobi equations, blowup argument and limiting generalized principal eigenvalue problems play critical roles.

Estimates for Sums of Eigenfunctions of Elliptic Pseudo-differential Operators on Compact Lie Groups

We extend the estimates proved by Donnelly and Fefferman, and by Lebeau, Robbiano and Zuazua, for sums of eigenfunctions of the Laplacian (on a compact manifold) to estimates for sums of eigenfunctions of any positive and elliptic pseudo-differential operator of positive order on a compact Lie group. Our criteria are imposed in terms of the positivity of the corresponding matrix-valued symbol of the operator. As an application of these inequalities in control theory, we obtain the null-controllability for diffusion models for elliptic pseudo-differential operators on compact Lie groups.

Elliptic fourth-order operators with Wentzell boundary conditions on Lipschitz domains

For bounded domains Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} with Lipschitz boundary Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}, we investigate boundary value problems for elliptic operators with variable coefficients of fourth-order subject to Wentzell (or dynamic) boundary conditions. Using form methods, we begin by showing general results for an even wider class of operators of type A=B∗B0-NbBγ,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathcal {A}}=\\begin{pmatrix} B^*B & 0 \\\\ -{\\mathscr {N}}^{\\mathfrak {b}}B & \\gamma \\end{pmatrix}, \\end{aligned}$$\\end{document}where B is associated to a quadratic form b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {b}}$$\\end{document} and Nb\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {N}}^{{\\mathfrak {b}}}$$\\end{document} an abstractly defined co-normal Neumann trace. Even in this general setting, we prove generation of an analytic semigroup on the product space H:=L2(Ω)×L2(Γ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {H}}:=L^2(\\Omega ) \ imes L^2(\\Gamma )$$\\end{document}. Using recent results concerning weak co-normal traces, we apply our abstract theory to the elliptic fourth-order case and are able to fully characterize the domain in terms of Sobolev regularity for operators in divergence form B=-divQ∇\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B=-\\mathop {{div} }Q \ abla $$\\end{document} with Q∈C1,1(Ω¯,Rd×d),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q \\in C^{1,1}({\\overline{\\Omega }},{\\mathbb {R}}^{d\ imes d}),$$\\end{document} also obtaining Hölder-regularity of solutions. Finally, we also discuss asymptotic behavior and (eventual) positivity.

A singular system involving mixed local and non-local operators

This article sets forth results on the existence, non-existence, uniqueness, and regularities properties, as well as boundary behavior of solutions for singular systems involving mixed local and non-local elliptic operators (see System (S) below). More precisely, we first establish a new weak comparison principle for a singular equation. Afterward, we discuss the non-existence of positive classical solutions, as well as construct suitable ordered pairs of sub-solutions and super-solutions. This allows us to obtain the existence of a pair of positive weak solutions for System (S) by employing Schauder’s fixed-point theorem in the associated conical shell. Finally, we adapt a method of Krasnoselsky to establish the uniqueness of such a positive pair of solutions.

Continuum limit for Laplace and elliptic operators on lattices

Continuum limit for Laplace and elliptic operators on lattices

Qualitative Analysis of General Elliptic Operator in Divergence Form

Qualitative Analysis of General Elliptic Operator in Divergence Form

Maximal $$L_p$$-regularity for x-dependent fractional heat equations with Dirichlet conditions

AbstractWe prove optimal regularity results in $$L_p$$ L p -based function spaces in space and time for a large class of linear parabolic equations with a nonlocal elliptic operator in bounded domains with limited smoothness. Here the nonlocal operator is given by a strongly elliptic and even pseudodifferential operator P of order 2a ($$0<a<1$$ 0 < a < 1 ) with nonsmooth x-dependent coefficients. This includes the prominent case of the fractional Laplacian $$(-\Delta )^a$$ ( - Δ ) a , as well as elliptic operators $$(-\nabla \cdot A(x)\nabla +b(x))^a$$ ( - ∇ · A ( x ) ∇ + b ( x ) ) a . The proofs are based on general results on maximal $$L_p$$ L p -regularity and its relation to $$\mathcal {R}$$ R -boundedness of the resolvent of the associated (elliptic) operator. Finally, we apply these results to show existence of strong solutions locally in time for a class of nonlinear nonlocal parabolic equations, which include a fractional nonlinear diffusion equation and a fractional porous medium equation after a transformation. The nonlinear results are new in the case of domains with boundary; the linear results are so when P is x-dependent nonsymmetric.

The logarithmic Dirichlet Laplacian on Ahlfors regular spaces

We introduce the logarithmic analogue of the Laplace-Beltrami operator on Ahlfors regular metric-measure spaces. This operator is intrinsically defined with spectral properties analogous to those of elliptic pseudo-differential operators on Riemannian manifolds. Specifically, its heat semigroup consists of compact operators which are trace-class after some critical point in time. Moreover, its domain is a Banach module over the Dini continuous functions and every Hölder continuous function is a smooth vector. Finally, the operator is compatible, in the sense of noncommutative geometry, with the action of a large class of non-isometric homeomorphisms.

Multiplicity results for nonlocal critical elliptic problems

AbstractWe prove new multiplicity results for some nonlocal critical growth elliptic problems in bounded domains. More specifically, we show that the problems considered here have arbitrarily many solutions for all sufficiently large values of a certain parameter . In particular, the number of solutions goes to infinity as . We also give an explicit lower bound on in order to have a given number of solutions. This lower bound is in terms of a sequence of eigenvalues of the associated nonlocal elliptic operator. The proofs are based on an abstract critical point theorem.

Index theorem for homogeneous spaces of Lie groups

We study index theory for homogeneous spaces associated to an almost connected Lie group in terms of the topological and analytic aspects. For the topological aspect, we obtain a topological formula as a result of the Riemann–Roch formula for proper cocompact actions of the Lie group, inspired by the work of Paradan and Vergne. For the analytic aspect, we apply heat kernel methods to obtain a local index formula representing the higher indices of equivariant elliptic operators with respect to a proper, cocompact action of the Lie group.

Deformation and $K$-theoretic index formulae on boundary groupoids

Motivated by investigating K -theoretic index formulae for boundary groupoids of the form {\mathcal{H}}= M_{0} \times M_{0} \sqcup G \times M_{1} \times M_{1} \rightrightarrows M = M_{0} \sqcup M_{1}, where G is an exponential Lie group, we introduce the notion of a deformation from the pair groupoid , which makes sense for general Lie groupoids. Once there exists a deformation from the pair groupoid for a (general) Lie groupoid {\mathcal{G}}\rightrightarrows M , we are able to construct explicitly a deformation index map relating the analytic index on {\mathcal{G}} and the index on the pair groupoid M \times M , which in turn enables us to establish index formulae for (fully) elliptic (pseudo)differential operators on {\mathcal{H}} by applying the numerical index formula of M. J. Pflaum, H. Posthuma, and X. Tang. In particular, we find that the index is given by the Atiyah–Singer integral but does not involve any \eta -term in the higher codimensional cases. These results recover and generalize our previous results for renormalizable boundary groupoids via renormalized traces.

Gaussian estimates vs. elliptic regularity on open sets

AbstractGiven an elliptic operator $$L= - {{\,\textrm{div}\,}}(A \nabla \cdot )$$ L = - div ( A ∇ · ) subject to mixed boundary conditions on an open subset of $$\mathbb {R}^d$$ R d , we study the relation between Gaussian pointwise estimates for the kernel of the associated heat semigroup, Hölder continuity of L-harmonic functions and the growth of the Dirichlet energy. To this end, we generalize an equivalence theorem of Auscher and Tchamitchian to the case of mixed boundary conditions and to open sets far beyond Lipschitz domains. Yet, we prove the consistency of our abstract result by encompassing operators with real-valued coefficients and their small complex perturbations into one of the aforementioned equivalent properties. The resulting kernel bounds open the door for developing a harmonic analysis for the associated semigroups on rough open sets.

Inverse initial‐value problems for time fractional diffusion equations in fractional Sobolev spaces

AbstractWe study the time fractional diffusion equation , , in a bounded domain with an elliptic operator and a locally Lipschitz nonlinearity on fractional Sobolev spaces, subjected to the homogeneous Dirichlet boundary condition. Data have not been measured at the initial time , but at a final time , that is, is given instead of . The problem is, therefore, called an inverse initial‐value problem. We first establish the well‐posedness of this problem on fractional Sobolev spaces and the regularity of the solution by assuming only the local Lipschitz continuity of . Second, an susceptible‐infected (shortly, SI) model with heterogeneity and a Navier–Stokes equation have been exemplified. Finally, a spatial ‐estimate for the solution and its gradient has been provided. The essential tools are asymptotic behaviours of Mittag–Leffler functions, fractional power spaces, fractional Sobolev spaces and embedding, weighted functional spaces, and estimates for heat semigroup.

Superlinear Krylov convergence under streamline diffusion FEM for convection‐dominated elliptic operators

AbstractThis paper studies the superlinear convergence of Krylov iterations under the streamline‐diffusion preconditioning operator for convection‐dominated elliptic problems. First, convergence results are given involving the diffusion parameter . Then the limiting case is studied on the operator level, and the convergence results are extended to this situation under some conditions, in spite of the lack of compactness of the perturbation operators. An explicit rate is also given.

Double forms: Regular elliptic bilaplacian operators

Double forms are sections of the vector bundles ΛkT∗M⊗ΛmT∗M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda^{k}T^{\\ast}{\\cal{M}}\\otimes\\Lambda^{m}T^{\\ast}\\cal{M}$$\\end{document}, where in this work (M,g\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\cal{M},\\frak{g}$$\\end{document}) is a compact Riemannian manifold with boundary. We study graded second-order differential operators on double forms, which are used in physical applications. A combination of these operators yields a fourth-order operator, which we call a double bilaplacian. We establish the regular ellipticity of the double bilaplacian for several sets of boundary conditions. Under additional conditions, we obtain a Hodge-like decomposition for double forms, whose components are images of the second-order operators, along with a biharmonic element. This analysis lays foundations for resolving several topics in incompatible elasticity, most prominently the existence of stress potentials and Saint-Venant compatibility.

Extrapolation of solvability of the regularity and the Poisson regularity problems in rough domains

Let Ω⊂Rn+1, n≥2, be an open set satisfying the corkscrew condition with n-Ahlfors regular boundary ∂Ω, but without any connectivity assumption. We study the connection between solvability of the regularity problem for divergence form elliptic operators with boundary data in the Hajłasz-Sobolev space M1,1(∂Ω) and the weak-A∞ property of the associated elliptic measure. In particular, we show that solvability of the regularity problem in M1,1(∂Ω) is equivalent to the solvability of the regularity problem in M1,p(∂Ω) for some p>1. We also prove analogous extrapolation results for the Poisson regularity problem defined on tent spaces. Moreover, under the hypothesis that ∂Ω supports a weak (1,1)-Poincaré inequality, we show that the solvability of the regularity problem in the Hajłasz-Sobolev space M1,1(∂Ω) is equivalent to a stronger solvability in a Hardy-Sobolev space of tangential derivatives.

Reiterated Homogenization of nonlinear degenerate elliptic operators with nonstandard growth

Reiterated Homogenization of nonlinear degenerate elliptic operators with nonstandard growth