The principal eigenvalue of a space–time periodic parabolic operator

This paper deals with the generalized principal eigenvalue of the parabolic operator , where the coefficients are periodic in t and x. We give the definition of this eigenvalue and we prove that it can be approximated by a sequence of principal eigenvalues associated to the same operator in a bounded domain, with periodicity in time and Dirichlet boundary conditions in space. Next, we define a family of periodic principal eigenvalues associated with the operator and use it to give a characterization of the generalized principal eigenvalue. Finally, we study the dependence of all these eigenvalues with respect to the coefficients.      

Linear and quasilinear parabolic equations in Sobolev space

We consider linear parabolic equations of second order in a Sobolev space setting. We obtain existence and uniqueness results for such equations on a closed two-dimensional manifold, with minimal assumptions about the regularity of the coefficients of the elliptic operator. In particular, we derive a priori estimates relating the Sobolev regularity of the coefficients of the elliptic operator to that of the solution. The results obtained are used in conjunction with an iteration argument to yield existence results for quasilinear parabolic equations.     

Parabolic equations with dynamical boundary conditions and source terms on interfaces

We consider parabolic equations with mixed boundary conditions and domain inhomogeneities supported on a lower dimensional hypersurface, enforcing a jump in the conormal derivative. Only minimal regularity assumptions on the domain and the coefficients are imposed. It is shown that the corresponding linear operator enjoys maximal parabolic regularity in a suitable \(L^p\) -setting. The linear results suffice to treat also the corresponding nondegenerate quasilinear problems.     

Maximum principle for quasilinear SPDE’s on a bounded domain without regularity assumptions

We prove a maximum principle for local solutions of quasi-linear parabolic stochastic PDEs, with non-homogeneous second order operator on a bounded domain and driven by a space–time white noise. Our method based on an approximation of the domain and the coefficients of the operator, does not require regularity assumptions. As in previous works by Denis et al. (2005, 2009)  and , the results are consequences of Itô’s formula and estimates for the positive part of local solutions which are non-positive on the lateral boundary.     

Elliptic operators and maximal regularity on periodic little-H?lder spaces

We consider one-dimensional inhomogeneous parabolic equations with higher-order elliptic differential operators subject to periodic boundary conditions. In our main result we show that the property of continuous maximal regularity is satisfied in the setting of periodic little-H?lder spaces, provided the coefficients of the differential operator satisfy minimal regularity assumptions. We address parameter-dependent elliptic equations, deriving invertibility and resolvent bounds which lead to results on generation of analytic semigroups. We also demonstrate that the techniques and results of the paper hold for elliptic differential operators with operator-valued coefficients, in the setting of vector-valued functions.     

Asymptotic behavior for the principal eigenvalue of a reinforcement problem

In this paper, we consider the asymptotic behavior for the principal eigenvalue of an elliptic operator with piecewise constant coefficients. This problem was first studied by Friedman in 1980. We show how the geometric shape of the interface affects the asymptotic behavior for the principal eigenvalue. This is a refinement of the result by Friedman.     

On the absence of rapidly decaying solutions for parabolic operators whose coefficients are non-Lipschitz continuous in time

We find minimal regularity conditions on the coefficients of a parabolic operator, ensuring that no nontrivial solution tends to zero faster than any exponential.

     

On a Class of Fully Nonlinear Parabolic Equations

We consider a class of intial boudnary value problems for parabolic equaitons of the form u$sub:t$esub:=f(t,x,u,Du,au) in a bounded domain Ω where A is an elliptic operator with continous coefficients. Scuh problems can be modeled by nonlinear evolation equaitons in Banach spaces, and we use abstract parabolic equairtions technique to show existence, uniqueness, regularity of a local solution, and to give sufficient conditions for existence in the large. In particular, we don't need growth assumptions on f with respect to Au to get existence in the large. In the case where Ω is a ball, A=D and f=f(t,|x||Du|², Du) we show that the solution is radially symmetric if the initial value is     

Backward uniqueness for parabolic operators whose coefficients are non-Lipschitz continuous in time

We investigate the relation between the backward uniqueness and the regularity of the coefficients for a parabolic operator. A necessary and sufficient condition for uniqueness is given in terms of the modulus of continuity of the coefficients.     

Positive stabilization of a parabolic equation by controls localized on a curve

We consider the stabilization of the nonnegative solutions of linear parabolic equation by controls localized on a curve. The main results of the article give a necessary and sufficient condition for positive stabilizability in terms of the principal eigenvalue of a certain elliptic operator. In case of positive stabilizability, some feedback stabilizing controls are indicated.     

Longtime behavior of the hyperbolic equations with an arbitrary internal damping

This paper is devoted to a study of the longtime behavior of the hyperbolic equations with an arbitrary internal damping, under sharp regularity assumptions that both the principal part coefficients and the boundary of the space domain (in which the system evolves) are continuously differentiable. For this purpose, we derive a new point-wise inequality for second differential operators with symmetric coefficients. Then, based on a global Carleman estimate, we establish an estimate on the underlying resolvent operator of the equation, via which, we show the logarithmic decay rate for solutions of the hyperbolic equations.     

Longtime behavior of the hyperbolic equations with an arbitrary internal damping

This paper is devoted to a study of the longtime behavior of the hyperbolic equations with an arbitrary internal damping, under sharp regularity assumptions that both the principal part coefficients and the boundary of the space domain (in which the system evolves) are continuously differentiable. For this purpose, we derive a new point-wise inequality for second differential operators with symmetric coefficients. Then, based on a global Carleman estimate, we establish an estimate on the underlying resolvent operator of the equation, via which, we show the logarithmic decay rate for solutions of the hyperbolic equations.     

Maximal Regular Abstract Elliptic Equations and Applications

The oblique derivative problem is addressed for an elliptic operator differential equation with variable coefficients in a smooth domain. Several conditions are obtained, guaranteing the maximal regularity, the Fredholm property, and the positivity of this problem in vector-valued L _p-spaces. The principal part of the corresponding differential operator is nonselfadjoint. We show the discreteness of the spectrum and completeness of the root elements of this differential operator. These results are applied to anisotropic elliptic equations.     

Boundary Properties of Solutions to Second-Order Parabolic Equations in Domains with Special Symmetry

We consider the first boundary-value problem for second-order nondivergent parabolic equations with, in general, discontinuous coefficients. We study the regularity of a boundary point assuming that in a neighborhood of this point the boundary of the domain is a surface of revolution. We prove a necessary and sufficient regularity condition in terms of parabolic capacities; for the heat equation this condition coincides with Wiener's criterion.     

Eigenvalue problems for the p-Laplacian

We study nonlinear eigenvalue problems for the p$p$-Laplace operator subject to different kinds of boundary conditions on a bounded domain. Using the Ljusternik–Schnirelman principle, we show the existence of a nondecreasing sequence of nonnegative eigenvalues. We prove the simplicity and isolation of the principal eigenvalue and give a characterization for the second eigenvalue.     

抛物型薛定谔方程的最优正则性估计(英文)

本文通过特征化抛物型薛定谔算子t-Δ+V的域,探讨了抛物型薛定谔方程u/t-Δu+Vu=f在Orlicz空间中的最优正则性估计,并且对空间的维数没有限制.     

Evolution systems for paraxial wave equations of Schrödinger-type with non-smooth coefficients

We prove existence of strongly continuous evolution systems in L² for Schrödinger-type equations with non-Lipschitz coefficients in the principal part. The underlying operator structure is motivated from models of paraxial approximations of wave propagation in geophysics. Thus, the evolution direction is a spatial coordinate (depth) with additional pseudodifferential terms in time and low regularity in the lateral space variables. We formulate and analyze the Cauchy problem in distribution spaces with mixed regularity. The key point in the evolution system construction is an elliptic regularity result, which enables us to precisely determine the common domain of the generators. The construction of a solution with low regularity in the coefficients is the basis for an inverse analysis which allows to infer the lack of lateral regularity in the medium from measured data.     

Perturbed preconditioned inverse iteration for operator eigenvalue problems with applications to adaptive wavelet discretization

In this paper we discuss an abstract iteration scheme for the calculation of the smallest eigenvalue of an elliptic operator eigenvalue problem. A short and geometric proof based on the preconditioned inverse iteration (PINVIT) for matrices (Knyazev and Neymeyr, SIAM J Matrix Anal 31:621–628, 2009) is extended to the case of operators. We show that convergence is retained up to any tolerance if one only uses approximate applications of operators which leads to the perturbed preconditioned inverse iteration (PPINVIT). We then analyze the Besov regularity of the eigenfunctions of the Poisson eigenvalue problem on a polygonal domain, showing the advantage of an adaptive solver to uniform refinement when using a stable wavelet base. A numerical example for PPINVIT, applied to the model problem on the L-shaped domain, is shown to reproduce the predicted behaviour.     

A nonlinear eigenvalue problem arising in a nanostructured quantum dot

In this paper we investigate a minimization problem related to the principal eigenvalue of the s-wave Schrödinger operator. The operator depends nonlinearly on the eigenparameter. We prove the existence of a solution for the optimization problem and the uniqueness will be addressed when the domain is a ball. The optimized solution can be applied to design new electronic and photonic devices based on the quantum dots.     

The asymptotic shift for the principal eigenvalue for second order elliptic operators in the presence of small obstacles

Let L be a uniformly elliptic second order differential operator with nice coefficients, defined on a smooth, bounded domain in ℝ ^d , d ≥ 2, with either the Dirichlet or an oblique-derivative boundary condition. In this work we study the asymptotics for the principal eigenvalue of L under hard and soft obstacle perturbations. The hard obstacle perturbation of L is obtained by making a finite number of holes with the Dirichlet boundary condition on their boundaries. The main result gives the asymptotic shift of the principal eigenvalue as the holes shrink to points. The rates are expressed in terms of the Newtonian capacity of the holes and the principal eigenfunctions for the unperturbed operator and its formal adjoint. The soft obstacle corresponds to a finite number of compactly supported finite potential wells. Here we only consider the oblique-derivative Laplacian. The main difference from the hard obstacle problem is that phase transitions occur, due to the various scaling possibilities. Our results generalize known results on similar perturbations for selfadjoint operators. Our approach is probabilistic.