Strichartz estimates for the wave equation on manifolds with boundary

We prove certain mixed-norm Strichartz estimates on manifolds with boundary. Using them we are able to prove new results for the critical and subcritical wave equation in 4-dimensions with Dirichlet or Neumann boundary conditions. We obtain global existence in the subcritical case, as well as global existence for the critical equation with small data. We also can use our Strichartz estimates to prove scattering results for the critical wave equation with Dirichlet boundary conditions in 3-dimensions.     

Boundary estimates for elliptic systems with L<Superscript>1</Superscript>–data

We obtain boundary estimates for the gradient of solutions to elliptic systems with Dirichlet or Neumann boundary conditions and L ¹–data, under some condition on the divergence of the data. Similar boundary estimates are obtained for div–curl and Hodge systems.     

Sous-solutions d'équations elliptiques dans L1

We study subsolutions for semilinear elliptic boundary value problems in L1. We consider as well nonlinear as linear boundary conditions. The nonlinear functions may be multivated. We characterize in terms of p.d.e. the subsolutions defined by a nonlinear functional analysis argument. Applications are given to obtain existence results for semilinear elliptic boundary value problems and comparison and estimates for nonlinear parabolic boundary value problems.     

Sobolev spaces of symmetric functions and applications

We prove sharp pointwise estimates for functions in the Sobolev spaces of radial functions defined in a ball. As a consequence, we obtain some imbeddings of such Sobolev spaces in weighted Lq-spaces. We also prove similar imbeddings for Sobolev spaces of functions with partial symmetry. Our techniques lead to new Hardy type inequalities. It is important to observe that we do not require any vanishing condition on the boundary to obtain all our estimates. We apply these imbeddings to obtain radial solutions and partially symmetric solutions for a biharmonic equation of the Hénon type under both Dirichlet and Navier boundary conditions. The delicate question of the regularity of these solutions is also established.     

Estimates of Hardy’s constants for tubular extensions of sets and domains with finite boundary moments

We study the behavior of constants in Hardy-type inequalities for tubular extensions of sets. We also obtain new estimates of constants for domains with finite boundary moments.     

On Some Properties of Orthogonal Polynomials over an Area in Domains of the Complex Plane. II

We investigate polynomials that are orthonormal with weight over the area of a domain with quasiconformal boundary. We obtain new exact estimates for the growth rate of these polynomials.     

Energy Decay for the Strongly Damped Nonlinear Beam Equation and Its Applications in Moving Boundary

We study the existence and energy decay of solutions for the strongly damped nonlinear beam equation. We apply a method based on Nakao method to show that the solution decays exponentially, and to obtain precise estimates of the constants in the estimates. Finally, we discuss its applications in moving boundary.     

Gradient Estimates in Fractional Dirichlet Problems

We obtain some fine gradient estimates near the boundary for solutions to fractional elliptic problems subject to exterior Dirichlet boundary conditions. Our results provide, in particular, the sign of the normal derivative of such solutions near the boundary of the underlying domain.

     

Lipschitz Estimates in Almost‐Periodic Homogenization

We establish uniform Lipschitz estimates for second‐order elliptic systems in divergence form with rapidly oscillating, almost‐periodic coefficients. We give interior estimates as well as estimates up to the boundary in bounded C^1,α domains with either Dirichlet or Neumann data. The main results extend those in the periodic setting due to Avellaneda and Lin for interior and Dirichlet boundary estimates and later Kenig, Lin, and Shen for the Neumann boundary conditions. In contrast to these papers, our arguments are constructive (and thus the constants are in principle computable) and the results for the Neumann conditions are new even in the periodic setting, since we can treat nonsymmetric coefficients. We also obtain uniform W^1,p estimates.© 2016 Wiley Periodicals, Inc.     

On the Solvability of the Hele–Shaw Model Problem in Weighted Hölder Spaces in a Plane Angle

We study a nonstationary boundary-value problem for the Laplace equation in a plane angle with time derivative in a boundary condition. We obtain coercive estimates in weighted Hölder spaces.     

Asymptotic Solution of Linear Bisingular Problems With Additional Boundary Layer

We study two bisingular Dirichlet problem with the additional boundary layer: 1) for the second order linear elliptic equation in a ring, 2) for linear ordinary differential equations of second order in a segment. We construct asymptotic solutions to the three-zone, bisingular Dirichlet problems by using the generalized method of boundary functions and obtain estimates for the residual functions.     

Square roots of elliptic second order divergence operators on strongly Lipschitz domains: $L^p$ theory

We study estimates for square roots of second order elliptic non necessarily selfadjoint operators in divergence form on Lipschitz domains subject to Dirichlet or to Neumann boundary conditions, pursuing our work [4] where we considered operators on . We obtain among other things for all if L is real symmetric and the domain bounded, which is new for . We also obtain similar results for perturbations of constant coefficients operators. Our methods rely on a singular integral representation, Calderón-Zygmund theory and quadratic estimates. A feature of this study is the use of a commutator between the resolvent of the Laplacian (Dirichlet and Neumann) and partial derivatives which carries the geometry of the boundary. Received: 12 January 2000 / Published online: 4 May 2001     

On quadrature and cubature formulas for a class of multiple singular integrals

We consider the problem of conditions for the existence of multiple singular integrals of a certain class at inner and boundary points of a domain. We obtain the quadrature and cubature formulas for calculating multiple singular integrals and present the corresponding estimates for the formulas.     

Boundary scalar controllability in projections for the Navier-Stokes system

We consider the following controllability problem for the Navier-Stokes equations: ensure, by dynamically varying the boundary values of the total fluid head, that the solution at a given time has a given projection onto some finite-dimensional subspace. We suggest a feedback control construction and obtain a priori estimates for the solution of the closed-loop system, which are then used to prove the boundary scalar controllability.     

Determination of coefficients for a dissipative wave equation via boundary measurements

In this paper we consider the inverse problem of recovering the viscosity coefficient in a dissipative wave equation via boundary measurements. We obtain stability estimates by considering all possible measurements implemented on the boundary. We also prove that the viscosity coefficient is uniquely determined by a finite number of measurements on the boundary provided that it belongs to a given finite dimensional vector space.     

On a problem with nonlinear boundary conditions for systems of functional-differential equations

We use the method of a priori estimates to obtain effective sufficient solvability conditions for systems of nonlinear functional-differential equations with nonlinear boundary conditions of periodic type. The results are in a sense optimal.     

Stability of Rothe difference scheme for the reverse parabolic problem with integral boundary condition

In this paper, we study the approximation of reverse parabolic problem with integral boundary condition. The Rothe difference scheme for an approximate solution of reverse problem is discussed. We establish stability and coercive stability estimates for the solution of the Rothe difference scheme. In sequel, we investigate the first order of accuracy difference scheme for approximation of boundary value problem for multidimensional reverse parabolic equation and obtain stability estimates for its solution. Finally, we give numerical results together with an explanation on the realization in one- and two-dimensional test examples.     

L 1 Hardy Inequalities with Weights

We prove sharp homogeneous improvements to L ¹ weighted Hardy inequalities involving distance from the boundary. In the case of a smooth domain, we obtain lower and upper estimates for the best constant of the remainder term. These estimates are sharp in the sense that they coincide when the domain is a ball or an infinite strip. In the case of a ball, we also obtain further improvements.     

Regularity for diffuse reflection boundary problem to the stationary linearized Boltzmann equation in a convex domain

We investigate the regularity issue for the diffuse reflection boundary problem to the stationary linearized Boltzmann equation for hard sphere potential, cutoff hard potential, or cutoff Maxwellian molecular gases in a strictly convex bounded domain. We obtain pointwise estimates for first derivatives of the solution provided the boundary temperature is bounded differentiable and the solution is bounded. This result can be understood as a stationary version of the velocity averaging lemma and mixture lemma.     

Uniform blow-up profiles and boundary layer for a parabolic system with nonlocal sources

This paper deals with blow-up properties of positive solutions to semi-linear parabolic system with nonlocal sources, subject to null Dirichlet boundary conditions. We first obtain some sufficient conditions for blow-up in finite time and a necessary condition for the two components of the solution blow up simultaneously, and then establish uniform blow-up profiles in the interior and estimates of the boundary layer.