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We develop a thorough mathematical analysis of the effective Mori–Zwanzig (EMZ) equation governing the dynamics of noise-averaged observables in stochastic differential equations driven by multiplicative Gaussian white noise. Building upon recent work on hypoelliptic operators, we prove that the EMZ memory kernel and fluctuation terms converge exponentially fast in time to a unique equilibrium state that admits an explicit representation. We apply the new theoretical results to the Langevin dynamics of a high-dimensional particle system with smooth interaction potential. [ABSTRACT FROM AUTHOR]

In this paper, we study the properties of a large class of zeta functions that arises in geometric analysis and mathematical physics. They are attached to some elliptic operators. This method can be used to evaluate explicitly the special values of zeta functions of elliptic operators defined on some symmetric spaces. [ABSTRACT FROM AUTHOR]

In this paper, we get two compactness results for complete manifolds by applying a (sub-) elliptic second-order differential operator on distance functions. The first is an extension of a theorem of Galloway and gets an upper estimate for the diameter of the manifold and the second is an extension of a theorem of Ambrose. [ABSTRACT FROM AUTHOR]

Let L=-div(A∇) be a second-order divergence form elliptic operator, where A is an accretive n×n matrix with bounded measurable complex coefficients in Rn. In this paper, we mainly establish the Lp boundedness for the commutators generated by b∈Iα(BMO) and the square function related to fractional differentiation for second-order elliptic operators. [ABSTRACT FROM AUTHOR]

We consider the decay of solution to a fractional diffusion equation with distributed order Caputo derivative. We assume that the elliptic operator is time-dependent and that the weight function, contained in the definition of the distributed order Caputo derivative, is just integrable. We establish the relation between behavior of the weight function near zero and the decay rate of solution. [ABSTRACT FROM AUTHOR]

Let X and Y be two noncommuting vector fields in an open set Ω in a manifold M equipped with a sub-Riemannian structure. We examine stable solutions of the following symmetric system Δ X Y u i = H i ( u 1 , ⋯ , u m ) in Ω for 1 ≤ i ≤ m , when the operator Δ X Y is the Hörmander's operator given by Δ X Y ( ⋅ ) : = X ( X ⋅ ) + Y ( Y ⋅ ) and H i ∈ C 1 ( R m ) . We prove the following identity for any w ∈ C 2 ( Ω ) | ∇ X Y X w | 2 + | ∇ X Y Y w | 2 − | X | ∇ X Y w | | 2 − | Y | ∇ X Y w | | 2 = { | ∇ X Y w | 2 [ A 2 + B 2 ] in { | ∇ X Y w | > 0 } ∩ Ω , 0 a . e . in { | ∇ X Y w | = 0 } ∩ Ω , where A is the intrinsic curvature of the level sets of w and B is connected with the intrinsic normal and the intrinsic tangent direction to the level sets and also with the Lie bracket [ X , Y ] . We then apply this to establish a geometric Poincaré inequality for stable solutions of the above system for general vector fields X and Y . This inequality enables us to analyze the level sets of stable solutions. In addition, we provide certain reduction of dimensions results which can be regarded as counterparts of the classical De Giorgi type results. This is remarkable since the classical one-dimensional symmetry results do not hold for general vector fields. Our approaches can be applied, but not limited, to the Grushin vector fields X = ( 1 , 0 ) and Y = ( 0 , x ) in R 2 and the Heisenberg vector fields X = ( 1 , 0 , − y 2 ) and Y = ( 0 , 1 , x 2 ) in R 3 and their multidimensional extensions. These specific vector fields generate nonelliptic operators which are hypoelliptic. [ABSTRACT FROM AUTHOR]

We study the problem of location of the spectrum of an elliptic operator in a bounded domain with both the classical and more special boundary conditions. On the complex plane, we construct the set of all eigenvalues of the corresponding operator and consider several examples to illustrate the “asymptotic accuracy” of the obtained set and consider an example of nonunique solvability of a parabolic inverse problem with final observation. [ABSTRACT FROM AUTHOR]

In this paper we present a new variational characteriztion of the first nontrival curve of the Fučík spectrum for elliptic operators with Dirichlet boundary conditions. Moreover, we describe the asymptotic behaviour and some properties of this curve and of the corresponding eigenfunctions. In particular, this new characterization allows us to compare the first curve of the Fučík spectrum with the infinitely many curves we obtained in previous works (see Molle and Passaseo, C R Math Acad Sci Paris 351:681-685, ; Ann I H Poincare-AN, ): for example, we show that these curves are all asymptotic to the same lines as the first curve, but they are all distinct from such a curve. [ABSTRACT FROM AUTHOR]

Conditions for well-posed and unique solvability of a non-homogeneous boundary value problem for a class of fourth order elliptic operator-differential equations with an unbounded operator in boundary conditions are found in this work. Note that these solvability conditions are sufficient, and they are expressed only in terms of the properties of operator coefficients of the boundary value problem. Besides, the estimates for the norms of intermediate derivative operators in a Sobolev-type space are obtained, and their close relationship with the solvability conditions is established. [ABSTRACT FROM AUTHOR]

We show that, for negative generators of analytic semigroups, a bounded [ABSTRACT FROM AUTHOR]

In this paper we aim to study the separation problem for the second order elliptic differential expression of the form E = ΔM + bi ð²/ ðxi² + q, with the real-valued positive continuous functions bi, on a complete Riemannian manifold (M, g) with metric g, where ΔM is the scalar Laplacian on M and q ≥ 0 is a locally square integrable function on M. In the terminology of Everitt and Giertz, E is said to be separated in L² (M) if for u ∈ L² (M) such that Eu ∈ L² (M), we have bi ð²/ ðxi² and qu ∈ L² (M). We give sufficient conditions for E to be separated in L² (M). [ABSTRACT FROM AUTHOR]

We affirmatively answer a question due to S. Böcherer concerning the feasibility of removing one differential operator from the standard collection of m + 1 of them used to embed the space of Jacobi forms of weight 2 and index m into several pieces of elliptic modular forms. [ABSTRACT FROM AUTHOR]

We deepen the study of the elliptic differential operator $$Au=\alpha u^{\prime \prime }+\beta u^{\prime }$$ on (weighted) spaces of continuous functions on a real interval. We establish several sufficient conditions implying, at the same time, the generation of positive $$C_{0}$$ -semigroups satisfying the Feller property and the existence of suitable cores. Some criteria on the regularity of the derivative, for functions satisfying Wentzell-type boundaries conditions, are also presented. [ABSTRACT FROM AUTHOR]

We consider elliptic boundary-value problems in which the elliptic operator is a polynomial function of a small parameter and the boundary conditions contain additional unknown functions. It is shown that the condition of ellipticity with small parameter is not only sufficient but also necessary for the a priori estimation of the solutions to this problem in the corresponding special functional spaces depending on the parameter. [ABSTRACT FROM AUTHOR]

This article presents an idea in the finite element methods (FEMs) for obtaining two-sided bounds of exact eigenvalues. This approach is based on the combination of nonconforming methods giving lower bounds of the eigenvalues and a postprocessing technique using conforming finite elements. Our results hold for the second and fourth-order problems defined on two-dimensional domains. First, we list analytic and experimental results concerning triangular and rectangular nonconforming elements which give at least asymptotically lower bounds of the exact eigenvalues. We present some new numerical experiments for the plate bending problem on a rectangular domain. The main result is that if we know an estimate from below by nonconforming FEM, then by using a postprocessing procedure we can obtain two-sided bounds of the first (essential) eigenvalue. For the other eigenvalues λ, l = 2, 3, ..., we prove and give conditions when this method is applicable. Finally, the numerical results presented and discussed in the paper illustrate the efficiency of our method. [ABSTRACT FROM AUTHOR]

Based on ideas of L. Alías, D. Impera and M. Rigoli developed in [13], we present a fairly general weak/Omori-Yau maximum principle for trace operators. We apply this version of maximum principle to generalize several higher order mean curvature estimates and to give an extension of Alias-Impera-Rigoli Slice Theorem of [13, Thm. 16 and 21], see Theorems 5 and 6. [ABSTRACT FROM AUTHOR]

We obtain a necessary and sufficient condition for the Hölder regularity of a boundary point O with respect to an elliptic second order differential operator. Sufficient conditions for the Hölder regularity of O are discussed. [ABSTRACT FROM AUTHOR]

Using an approach based on the heat kernel, we prove an Atiyah–Bott–Lefschetz theorem for the $$L^2$$ L 2 -Lefschetz numbers associated with an elliptic complex of cone differential operators over a compact manifold with conical singularities. We then apply our results to the case of the de Rham complex. [ABSTRACT FROM AUTHOR]

We obtain a constructive description of all Hilbert function spaces that are interpolation spaces with respect to a pair of Sobolev spaces $$ \left[ {{H^{{\left( {{s_0}} \right)}}}\left( {{{\mathbb{R}}^n}} \right),{H^{{\left( {{s_1}} \right)}}}\left( {{{\mathbb{R}}^n}} \right)} \right] $$ of some integer orders s and s and form an extended Sobolev scale. We propose equivalent definitions of these spaces with the use of uniformly elliptic pseudo-differential operators positive-definite in $$ {L_2}\left( {{{\mathbb{R}}^n}} \right) $$. Possible applications of the introduced scale of spaces are indicated. [ABSTRACT FROM AUTHOR]

Abstract: We consider a sequence of linear Dirichlet problems as follows with uniformly elliptic and possibly non-symmetric. Using purely variational arguments we give an alternative proof of the compactness of H-convergence, originally proved by Murat and Tartar. [Copyright &y& Elsevier]

Abstract: In this paper, we establish the existence of two nontrivial solutions to a class of nonlocal hemivariational inequalities depending on two parameters. Our methods are based on critical point theory for non-differentiable functionals. [Copyright &y& Elsevier]

We prove the existence of three weak solutions of a quasilinear elliptic system involving a general (p; q)-elliptic operator in divergence form, with 1 < p ≤ n, 1 < q ≤ n. Our main tool is an adaptation of a three critical points theorem due to Ricceri. [ABSTRACT FROM AUTHOR]

In this article, we consider the nonautonomous evolution problem Du/dt = a(t)Au(t); 0 ≤ s ≤ t < T with initial condition u(s) = χ where -A generates a holomorphic semigroup of angle ϑ ∈ (0, π/2] on a Banach space X and a ∈ C([0, T] : ℝ+). The problem is generally ill-posed under such conditions, and so we employ methods to approximate known solutions of the problem. In particular, we prove the existence of a family of regularizing operators for the problem which stems from the solution of an approximate well-posed problem. In fact, depending on whether ϑ ∈ (0, π/4] or ϑ 2 (π/4; π/2], we provide two separate approximations each yielding a regularizing family. The theory has applications to ill-posed partial differential equations in Lp(Ω), 1 < p < 1 where A is a strongly elliptic differential operator and Ω is a fixed domain in ℝn. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we extend a recently established three-critical-point theorem of Arcoya and Carmona (2007)  relating to some functionals which are only differentiable along some directions to Motreanu–Panagiotopoulos functionals. As its applications, we establish the existence of two nontrivial solutions to hemivariational inequalities driven by non-local integrodifferential operators. [Copyright &y& Elsevier]

We show that (1) if A is a nonzero quasinilpotent operator with ran A n closed for some n ≥ 1, then its numerical range W(A) contains 0 in its interior and has a differentiable boundary, and (2) a noncircular elliptic disc can be the numerical range of a nilpotent operator with nilpotency 3 on an infinite-dimensional separable space. (1) is a generalization of the known result for nonzero nilpotent operators, and (2) is in contrast to the finite-dimensional case, where the only elliptic discs which are the numerical ranges of nilpotent finite matrices are the circular ones centred at the origin. [ABSTRACT FROM AUTHOR]

We study the difference between vertical and conical square functions in the abstract and also in the specific case where the square functions come from an elliptic operator. [ABSTRACT FROM AUTHOR]

We prove bordism invariance of the coarse index of complex elliptic pseudodifferential operators. In our discussion we introduce directed c-bordisms, whose usefulness is illustrated in the context of existence of uniformly positive scalar curvature metrics on open manifolds. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the authors first show that the classical Hardy space can be characterized by the non-tangential maximal functions and the area integrals associated with the semigroups and , respectively, where is an elliptic operator with real constant coefficients of homogeneous order 2m (). Moreover, the authors also prove that can be characterized by the Riesz transforms if and only if m is an odd integer. In the main part of this paper, the authors develop a theory of Hardy space associated with L, where L is a higher order divergence form elliptic operator with complex bounded measurable coefficients. The authors set up a molecular Hardy space and give its characterizations by area integrals related to the semigroups and , respectively. Finally, authors give the boundedness of Riesz transforms, square functions and maximal functions associated with L. [Copyright &y& Elsevier]

We prove the Kato conjecture for degenerate elliptic operators on Rn. More precisely, we consider the divergence form operator Lw = -w-1 div A ∇, where w is a Muckenhoupt A2 weight and A is a complex-valued n Χ n matrix such that w-1 A is bounded and uniformly elliptic. We show that if the heat kernel of the associated semigroup e--tLw satisfies Gaussian bounds, then the weighted Kato square root estimate, ‖L1/2 w f‖L2(w) ≈ ‖∇f‖L2(w), holds. [ABSTRACT FROM AUTHOR]

The paper deals with a Dirichlet spectral problem for an elliptic operator with ε-periodic coefficients in a 3D bounded domain of small thickness δ. We study the asymptotic behavior of the spectrum as ε and δ tend to zero. This asymptotic behavior depends crucially on whether ε and δ are of the same order (δ ≈ ε), or ε is much less than δ(δ = ετ, τ < 1), or ε is much greater than δ(δ = ετ, τ > 1). We consider all three cases. [ABSTRACT FROM PUBLISHER]

The article discusses Gårding's inequality with nonpower degeneration for higher order elliptic operators. It says that from the partial differential equations' general theory, the Gårding's inequality has an important role in the uniformly elliptic equations of boundary value problems' solvability analysis. It presents an equation that does not use Gårding's inequality, however, it analyzes the degenerate elliptic equations of the Dirichlet problem. Furthermore, it states that the elliptic equation with degeneration was solved using Gårding's inequality, wherein its elliptic operators were considered as special because it only degenerate on one independent variable.

It is proved that the moduli space of all connected compact orientable embedded minimal affine Lagrangian submanifolds of a complex equiaffine space constitutes an infinite dimensional Fréchet manifold (if it is not Ø). [ABSTRACT FROM AUTHOR]

Abstract: In this paper we prove Hessian and Laplacian comparison theorems for the Lorentzian distance function in a spacetime with sectional (or Ricci) curvature bounded by a certain function by means of a comparison criterion for Riccati equations. Using these results, under suitable conditions, we are able to obtain some estimates on the higher order mean curvatures of spacelike hypersurfaces satisfying a Omori–Yau maximum principle for certain elliptic operators. [Copyright &y& Elsevier]

Abstract: For selfadjoint extensions of a symmetric densely defined positive operator , the lower boundedness problem is the question of whether is lower bounded if and only if an associated operator T in abstract boundary spaces is lower bounded. It holds when the Friedrichs extension has compact inverse (Grubb, 1974, also Gorbachuk and Mikhailets, 1976); this applies to elliptic operators A on bounded domains. For exterior domains, is not compact, and whereas the lower bounds satisfy , the implication of lower boundedness from T to has only been known when . We now show it for general T. The operator corresponding to , generalizing the Krein–von Neumann extension , appears here; its possible lower boundedness for all real a is decisive. We study this Krein-like extension, showing for bounded domains that the discrete eigenvalues satisfy for . [Copyright &y& Elsevier]

We study the behavior of Fourier integrals summed by the symbols of elliptic operators and pointwise convergence of Fourier inversion. We consider generalized localization principle which in classical Lp spaces was investigated by Sjölin (1983), Carbery and Soria (1988, 1997) and Alimov (1993). Proceeding these studies, in this paper, we establish sharp conditions for generalized localization in the class of finitely supported distributions. [ABSTRACT FROM AUTHOR]

In this work we prove the Aleksandrov-Bakel'man-Pucci estimate for (possibly degenerate) nonlinear elliptic and parabolic equations of the form and for F a $${\fancyscript{C}^1}$$ monotone field under some suitable conditions. Examples of applications such as the p-Laplacian and the Mean Curvature Flow are considered, as well as extensions of the general results to equations that are not in divergence form, such as the m-curvature flow. [ABSTRACT FROM AUTHOR]

We investigate the Dirichlet weighted eigenvalue problem for a fourth-order elliptic operator with variable coefficients in a bounded domain in $ {\mathbb{R}^n} $. We establish a sharp inequality for its eigenvalues. It yields an estimate for the upper bound of the ( k + 1)th eigenvalue in terms of the first k eigenvalues. Moreover, we also obtain estimates for some special cases of this problem. In particular, our results generalize the Wang-Xia inequality ( J. Funct. Anal., 245, No. 1, 334-352 ()) for the clamped-plate problem to a fourth-order elliptic operator with variable coefficients. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to extend in Rn a result of Seeley on eigenfunction expansions of real analytic functions on compact manifolds. As a counterpart of an elliptic operator in a compact manifold, we consider in Rn a selfadjoint, globally elliptic Shubin type differential operator with spectrum consisting of a sequence of eigenvalues λj, j ϵ N, and a corresponding sequence of eigenfunctions uj, j ϵ N, forming an orthonormal basis of L²(Rn). Elements of Schwartz S(Rn), resp. Gelfand-Shilov S½½ spaces, are characterized through expansions Σj ajuj and the estimates of coefficients aj by the power function, resp. exponential function of λj . [ABSTRACT FROM AUTHOR]

The main goal of this paper is to extend in Rn a result of Seeley on eigenfunction expansions of real analytic functions on compact manifolds. As a counterpart of an elliptic operator in a compact manifold, we consider in Rn a selfadjoint, globally elliptic Shubin type differential operator with spectrum consisting of a sequence of eigenvalues λj, j ϵ N, and a corresponding sequence of eigenfunctions uj, j ϵ N, forming an orthonormal basis of L²(Rn). Elements of Schwartz S(Rn), resp. Gelfand-Shilov S½½ spaces, are characterized through expansions Σj ajuj and the estimates of coefficients aj by the power function, resp. exponential function of λj . [ABSTRACT FROM AUTHOR]

We survey recent results related to uniqueness problems for parabolic equations for measures. We consider equations of the form ∂ μ = L μ for bounded Borel measures on ℝ × (0, T), where L is a second order elliptic operator, for example, $$ Lu = {\Delta_x}u + \left( {b,{\nabla_x}u} \right) $$, and the equation is understood as the identityfor all smooth functions u with compact support in ℝ × (0, T). Our study are motivated by equations of such a type, namely, the Fokker-Planck-Kolmogorov equations for transition probabilities of diffusion processes. Solutions are considered in the class of probability measures and in the class of signed measures with integrable densities. We present some recent positive results, give counterexamples, and formulate open problems. Bibliography: 34 titles. [ABSTRACT FROM AUTHOR]

In this paper we investigate properties of solutions of first and second order elliptic equations that degenerate along a simple closed curve in $ \mathbb{R}^2$ $ \mathbb{C}$. To the vector field $ L$ $ \mathbb{P}=\mathrm{Re}\left[L\overline{L} L \right]$ is a $ \mathbb{C}$ $ \mathbb{P}u=0$ $ Lw=Aw\overline{w}$ [ABSTRACT FROM AUTHOR]

Abstract: Picone identities are established for a class of half-linear elliptic operators with -Laplacians, and Sturmian comparison theorems are obtained on the basis of the Picone identities. Specializations to half-linear partial or ordinary differential inequalities with -Laplacians are shown. [Copyright &y& Elsevier]

We consider a nonlinear Neumann problem driven by a nonhomogeneous quasilinear degenerate elliptic differential operator $ \operatorname{div} a(x,\nabla u)$-Laplacian. The reaction term is a Carathéodory function $ f(x,s)$. Using variational methods based on the mountain pass and second deformation theorems, together with truncation and minimization techniques, we show that the problem has three nontrivial smooth solutions, two of which have constant sign (one positive, the other negative). A crucial tool in our analysis is a result of independent interest which we prove here and which relates $ W^{1,p}$ local minimizers of a $ C^1$ $ \operatorname{div} a(x,\nabla u)$ [ABSTRACT FROM AUTHOR]

Abstract: We establish Dahlbergʼs perturbation theorem for non-divergence form operators . If and are two operators on a Lipschitz domain such that the Dirichlet problem for the operator is solvable for some and the coefficients of the two operators are sufficiently close in the sense of Carleson measure, then the Dirichlet problem for the operator is solvable for the same p. This is a refinement of the version of this result proved by Rios (2003) in . As a consequence we also improve a result from Dindoš et al. (2007) for the solvability of non-divergence form operators (Theorem 3.2) by substantially weakening the condition required on the coefficients of the operator. The improved condition is exactly the same one as is required for divergence form operators . [Copyright &y& Elsevier]

We give a method of representing fully nonlinear elliptic operators given by boundedly inhomogeneous functions in terms of operators acting only on pure second order derivatives. We also discuss possible applications of such representations to using .finite di.erence approximations for solving the corresponding equations. [ABSTRACT FROM AUTHOR]

For a second order differential operator $ \mathcal{A} $ = −div g( x/ ε)∇ + εp( x/ ε) in L(ℝ) with periodic coefficients and small parameter ε > 0 we study an approximation of the resolvent of $ \mathcal{A} $ at a point close to an edge of an inner gap in the spectrum of $ \mathcal{A} $. Under certain regularity conditions, we construct an approximation (with a first order corrector taken into account) for the resolvent with error estimate of order O( ε). We show that a proper effective operator and a proper corrector are associated to each (regular) edge of the gap. Bibliography: 14 titles. [ABSTRACT FROM AUTHOR]

In the class of holomorphic vector functions, we determine the conditions of solvability of the boundaryvalue problem for a class of second-order operator differential equations expressed in terms of the operator coefficients appearing both in the equation and in the boundary condition. [ABSTRACT FROM AUTHOR]

The article presents a methodology for calculating nonlinear elliptic equations for probability measurement. It provides the second-order elliptic operator in which the summation over repeated indices is expressed, and offers an alternative of the operator which can be written in coordinate-free form. It introduces a theorem for a solution of elliptic equation under assumptions on A and b for operators with coefficients not depending on μ. It adds that an adequate condition for such theorem is the presence of Lyapunov functions.