Singular perturbation degenerate problems occurring in atmospheric dispersion of pollutants

The boundary value problems for the degenerate differential-operator equations with small parameters generated on all boundary are studied. Several conditions for the separability and the fredholmness in Banach-valued L_p-spaces of are given. In applications, maximal regularity of degenerate Cauchy problem for parabolic equation arising in atmospheric dispersion of pollutants studied.     

A Linearization Method for Nondegenerate Variational Conditions

We employ recent results about constraint nondegeneracy in variational conditions to design and justify a linearization algorithm for solving such problems. The algorithm solves a sequence of affine variational inequalities, but the variational condition itself need not be a variational inequality: that is, its underlying set need not be convex. However, that set must be given by systems of differentiable nonlinear equations with additional polyhedral constraints. We show that if the variational condition has a solution satisfying nondegeneracy and a standard regularity condition, and if the linearization algorithm is started sufficiently close to that solution, the algorithm will produce a well defined sequence that converges Q-superlinearly to the solution.     

Anti-self-dual Lagrangians: Variational resolutions of non-self-adjoint equations and dissipative evolutions

We develop the concept and the calculus of anti-self-dual (ASD) Lagrangians and their derived vector fields which seem inherent to many partial differential equations and evolutionary systems. They are natural extensions of gradients of convex functions – hence of self-adjoint positive operators – which usually drive dissipative systems, but also provide representations for the superposition of such gradients with skew-symmetric operators which normally generate unitary flows. They yield variational formulations and resolutions for large classes of non-potential boundary value problems and initial-value parabolic equations. Solutions are minima of newly devised energy functionals, however, and just like the self (and anti-self) dual equations of quantum field theory (e.g. Yang–Mills) the equations associated to such minima are not derived from the fact they are critical points of the functional I, but because they are also zeroes of suitably derived Lagrangians. The approach has many advantages: it solves variationally many equations and systems that cannot be obtained as Euler–Lagrange equations of action functionals, since they can involve non-self-adjoint or other non-potential operators; it also associates variational principles to variational inequalities, and to various dissipative initial-value first order parabolic problems. These equations can therefore be analyzed with the full range of methods – computational or not – that are available for variational settings. Most remarkable are the permanence properties that ASD Lagrangians possess making their calculus relatively manageable and their domain of applications quite broad.     

带梯度障碍的抛物型变分不等式解的存在性唯一性和正则性

本文讨论带梯度障碍的抛物型变分不等式解的存在性、唯一性和正则性问题．通过证明一类带梯度障碍的问题的求解等价于解某个双边障碍的问题，并利用双边障碍问题解的存在性、唯一性和正则性，得到了带梯度障碍的问题的相应结果．这一方法将有助于对具有梯度约束的非线性以及完全非线性抛物型方程解的正则性的研究．     

Maximal parabolic regularity for divergence operators including mixed boundary conditions

We show that elliptic second order operators A of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth, the coefficients of A are discontinuous and A is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.     

Trapping regions for discontinuously coupled systems of evolution variational inequalities and application

We consider initial-boundary value problems for weakly coupled systems of parabolic equations under coupled nonlinear flux boundary condition. Both coupling vector fields and are assumed to be either of competitive or cooperative type, but may otherwise be discontinuous with respect to all their arguments. The main goal is to provide conditions for the vector fields f and g that allow the identification of regions of existence of solutions (so called trapping regions). To this end the problem is transformed to a discontinuously coupled system of evolution variational inequalities. Assuming a generalized outward pointing vector field on the boundary of a rectangle of the dependent variable space, the system of evolution variational inequalities is solved via a fixed point problem for some increasing operator in an appropriate ordered Banach space. The main tools used in the proof are evolution variational inequalities, comparison techniques, and fixed point results in ordered Banach spaces.     

BOUNDARY ELEMENT APPROXIMATION OF THE SEMI-DISCRETE PARABOLIC VARIATIONAL INEQUALITIES OF THE SECOND KIND

The boundary element approximation of the parabolic variational inequalities of the second kind is discussed. First, the parabolic variational inequalities of the second kind can be reduced to an elliptic variational inequality by using semidiscretization and implicit method in time; then the existence and uniqueness for the solution of nonlinear non-differentiable mixed variational inequality is discussed. Its corresponding mixed boundary variational inequality and the existence and uniqueness of its solution are yielded. This provides the theoretical basis for using boundary element method to solve the mixed vuriational inequality.     

The Laplacian on Cylindrical Domains

The aim of this paper is to prove elliptic regularity and parabolic maximal regularity of the Laplacian with mixed boundary conditions on domains Ω carrying a cylindrical structure. More precisely, we consider Ω to be given as the Cartesian product of whole or half spaces, a cube ${\mathcal{Q}}$ , and a standard domain V having compact boundary. Taking advantage of this structure we apply operator-valued Fourier multiplier results to transfer ${\mathcal{H}^{\infty}}$ -calculus results known for the Laplacian in L ^p (V) to the Laplacian in L ^p (Ω). This approach turns out to inherit elliptic regularity, i.e. the domain of the Dirichlet Laplacian equals ${W^{2,p}(\Omega) \cap W_0^{1,p}(\Omega)}$ , for instance. This is surprising since Ω may be unbounded and non-convex with boundary neither compact nor of class C ^1,1 at the same time. More generally, we consider the following mixture of boundary conditions: on every smooth part of the boundary Dirichlet or Neumann boundary conditions are imposed and on parts related to ${\mathcal{Q}}$ generalized periodic boundary conditions are included. Via ${\mathcal{R}}$ -sectoriality we deduce maximal regularity in the parabolic sense which seems to be new for this general class of boundary conditions. Parabolic equations with such a mixture of boundary conditions on such type of domains appear for example in models describing growth of biological cells.     

One-Dimensional Numerical Algorithms for Gradient Flows in the p-Wasserstein Spaces

We numerically approximate, on the real line, solutions to a large class of parabolic partial differential equations which are “gradient flows” of some energy functionals with respect to the L ^p -Wasserstein metrics for all p>1. Our method relies on variational principles involving the optimal transport problem with general strictly convex cost functions.     

Separable anisotropic elliptic operators and applications

This paper focuses on boundary value problems for anisotropic differential-operator equations of high order with variable coefficients in the half plane. Several conditions are obtained which guarantee the maximal regularity of anisotropic elliptic and parabolic problems in Banach-valued L _p -spaces. Especially, it is shown that this differential operator is R-positive and is a generator of an analytic semigroup. These results are also applied to infinite systems of anisotropic type partial differential equations in the half plane.     

Orlicz regularity for higher order parabolic equations in divergence form with coefficients in weak BMO

We consider higher order parabolic equations in divergence form with measurable coefficients to find optimal regularity in Orlicz spaces of the maximum order derivatives of the weak solutions. The relevant minimal regularity requirement on the tensor matrix coefficients is of small BMO in the spatial variable and is measurable in the time variable. As a consequence we prove the classical W ^m,p regularity, m = 1, 2, . . . , 1 < p < ∞, for such higher order equations. In the same spirit the results easily extend to higher order parabolic systems as well as up to the boundary.     

On the p-version approximation in the boundary element method for a variational inequality of the second kind modelling unilateral contact and given friction

In this paper we complement recent work of Maischak and Stephan on adaptive hp-versions of the BEM for unilateral Signorini problems, respectively on FEM-BEM coupling in its h-version for a nonlinear transmission problem modelling Coulomb friction contact. Here we focus on the boundary element method in its p-version to treat a scalar variational inequality of the second kind that models unilateral contact and Coulomb friction in elasticity together. This leads to a nonconforming discretization scheme. In contrast to the work cited above and to a related paper of Guediri on a boundary variational inequality of the second kind modelling friction we take the quadrature error of the friction functional into account of the error analysis. At first without any regularity assumptions, we prove convergence of the BEM Galerkin approximation in the energy norm. Then under mild regularity assumptions, we establish an a priori error estimate that is based on a novel Céa–Falk lemma for abstract variational inequalities of the second kind.     

Nonuniformly nonlinear elliptic equations of p-biharmonic type

This paper deals with the existence and multiplicity of weak solutions to nonlinear differential equations involving a general p-biharmonic operator (in particular, p-biharmonic operator) under Dirichlet boundary conditions or Navier boundary conditions. Our method is mainly based on variational arguments.     

On quasilinear parabolic evolution equations in weighted L p -spaces II

Our study of abstract quasi-linear parabolic problems in time-weighted L _p -spaces, begun in Köhne et al. (J Evol Equ 10:443–463, 2010), is extended in this paper to include singular lower-order terms, while keeping low initial regularity. The results are applied to reaction-diffusion problems, including Maxwell–Stefan diffusion, and to geometric evolution equations like the surface diffusion flow or the Willmore flow. The method presented here will be applicable to other parabolic systems, including free boundary problems.     

Optimal regularity estimates for parabolic p-harmonic equations

In this paper optimal regularity estimates for weak solutions of quasilinear parabolic equations of p-Laplacian type with small BMO coefficients are investigated. Our results improve the known results for such equations using a harmonic analysis free technique.     

Maximal regularity of type L p for abstract parabolic Volterra equations

We prove maximal regularity results of type L_p for abstract parabolic Volterra equations including problems with inhomogeneous boundary data. Our approach is purely operator theoretic. It uses the inversion of the convolution, the Dore-Venni theorem, the Mikhlin theorem in the operator-valued version, and real interpolation. Known results on L_p-regularity of abstract Cauchy problems and abstract parabolic pdes with inhomogeneous boundary conditions are recovered. As an application we consider the heat equation of memory type with inhomogeneous boundary condition.     

On nonlinear parabolic variational inequalities containing nonlocal terms

We investigate nonlinear parabolic variational inequalities which contain functional dependence on the unknown function. Such parabolic functional differential equations were studied e.g. by L. Simon in [8] (which was motivated by the work of M. Chipot and L. Molinet in [4]), where the following equation was considered: (1) $$ \begin{array}{*{20}c} {D_t u(t,x) - \sum\limits_{i = 1}^n {D_i \left[ {a_i (t,x,u(t,x),Du(t,x);u)} \right]} } \\ { + a_0 (t,x,u(t,x),Du(t,x);u) = f(t,x)} \\ {(t,x) \in Q_T = (0,T) \times \Omega ,a_i :Q_T \times R^{n + 1} \times L^p (0,T;V) \to R,} \\ \end{array} $$ where V denotes a closed linear subspace of the Sobolev-space W ^1,p (Ω) (2 ≦ p < ∞). In the above mentioned paper existence of weak solutions of the above equation is shown. These results were extended to systems of functional differential equations in [2]. In the following, we extend these existence results to variational inequalities by using the (less known) results of [6]. Finally, we show some examples.     

Coercive boundary value problems for regular degenerate differential-operator equations

This study focuses on nonlocal boundary value problems (BVP) for degenerate elliptic differential-operator equations (DOE), that are defined in Banach-valued function spaces, where boundary conditions contain a degenerate function and a principal part of the equation possess varying coefficients. Several conditions obtained, that guarantee the maximal L_p regularity and Fredholmness. These results are also applied to nonlocal BVP for regular degenerate partial differential equations on cylindrical domain to obtain the algebraic conditions that ensure the same properties.     

On mixed problem for parabolic variational inequalities

Summary We give an existence-uniqueness result for the parabolic variational inequality (1.3) and we apply this result to mixed problem for parabolic variational inequalities and to parabolic quasi-variational inequalities with constraints on the boundary.

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Sunto. Si dà un risultato di esistenza ed unicità per la disequazione variazionale parabolica (1.3) e si applica tale risultato al problema misto per disequazioni variazionali paraboliche e a disequazioni quasi variazionali paraboliche con vincolo sulla frontiera.

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Entrata in Redazione il 7 giugno 1976.

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Politecnico di Milano, Istituto di Matematica (Analisi); lavoro eseguito nell'ambito del gruppo G.N.A.F.A. del C.N.R.     

Optimal Control of a Parabolic Equation with Dynamic Boundary Condition

We investigate a control problem for the heat equation. The goal is to find an optimal heat transfer coefficient in the dynamic boundary condition such that a desired temperature distribution at the boundary is adhered. To this end we consider a function space setting in which the heat flux across the boundary is forced to be an L ^p function with respect to the surface measure, which in turn implies higher regularity for the time derivative of temperature. We show that the corresponding elliptic operator generates a strongly continuous semigroup of contractions and apply the concept of maximal parabolic regularity. This allows to show the existence of an optimal control and the derivation of necessary and sufficient optimality conditions.