Nonlinear artificial boundary conditions with pointwise error estimates for the exterior three dimensional Navier–Stokes problem

On a three–dimensional exterior domain Ω we consider the Dirichlet problem for the stationary Navier–Stokes system. We construct an approximation problem on the domain Ω_R, which is the intersection of Ω with a sufficiently large ball, while we create nonlinear, but local artificial boundary conditions on the truncation boundary. We prove existence and uniqueness of the solutions to the approximating problem together with asymptotically precise pointwise error estimates as R tends to infinity.     

Partial regularity for a minimum problem with free boundary

Regularity of the free boundary ?{u > 0} of a non-negative minimum u of the functional $\upsilon \mapsto \int\limits_\Omega {\left( {\left| {\nabla \upsilon } \right|^2 + Q^2 \chi _{\left\{ {\upsilon > 0} \right\}} } \right)} $ , where Ω is an open set in ?ⁿ and Q is a strictly positive Hölder-continuous function, is still an open problem for n ≥ 3. By means of a new monotonicity formula we prove that the existence of singularities is equivalent to the existence of an absolute minimum u* such that the graph of u* is a cone with vertex at 0, the free boundary ?{u* > 0} has one and only one singularity, and the set {u* > 0} minimizes the perimeter among all its subsets. This leads to the following partial regularity: there is a maximal dimension k* ≥ 3 such that for n < k* the free boundary ?{u > 0} is locally in Ω a C^1,α-surface, for n = k* the singular set Σ:= ?{u > 0} ? ?_red{u > 0} consists at most of in Ω isolated points, and for n > k* the Hausdorff dimension of the singular set Σ is less than n - k*.     

Geometry of unbounded domains,poincaré inequalities and stability in semilinear parabolic equations

We investigate thc close relations existing between certain geometric properties of domains Ω of R^N, the validity of Poincark inequalities in Ω, and the behavior of solutions of semilinear parabolic equations. For the equation u_t-△u=|u|^p-1 we obtain a purely geometric, necessary and sufficient condition on Ω, for the 0 solution to be asymptotically (and exponentially) stable in L^r(ω)1<r<∞ when r is supercritical(r>N(p-1)/2 . The condition is that the inradius of Ω be finite. The result is different for r critical. For the equation u_t-△u=u^p-μ|u|^q,q≥p>1,μ>0 we prove that the finiteness of the inradius is a necessary and sufficient condition for global existence and boundedness of all nonnegative solutions.     

Existence Results for Classes of Sublinear Semipositone Problems

We consider the semipositone problem $${\matrix {-\Delta u (x)= \lambda f (u(x))\ \ \; \ \ \ \ \ x \in \Omega \cr \qquad \qquad \qquad u(x)=0 \ \ \ \;\ \ \ \ x \in \partial \Omega \cr}}$$ where λ > 0 is a constant, Ω is a bounded region in Rⁿ with a smooth boundary, and f is a smooth function such that f ′(u) is bounded below, f (0) < 0 and \({\rm lim}_{u \rightarrow}+\infty {f(u)\over u}=0. \) We prove under some additional conditions the existence of a positive solution (1) for λ ∈ I where I is an interval close to the smallest eigenvalue of —Δ with Dirichlet boundary condition and (2) for λ large. We also prove that our solution u for λ large is such that∥u∥ ? sup_x∈Ω ¦u(x)¦ → ∞ as A → ∞. Our methods are based on sub and super solutions. In particular, we use an anti maximum principle to obtain a subsolution for our existence result for λ ∈ I.     

Stability For a Multidimensional Inverse Spectral Theorem

We consider the eigenvalue problem in Ω

Where Ω is a bounded domain in R^d with smooth boundary,a nd q is a bounded, measurable function on Ω The eigenvalue problem has discrete spectrum; we denote by and a nondecreasing sequence of eigenvalue and corresponding (orthonormal) eigenfunctions. It is known ([N–S–U]) that knowledge of the eigenvalues and the boundary values of the normal derivatives of the corresponding eigenfunctions is sufficient to uniquely determine a coefficient, q.     

Shape optimization for Stokes problem with threshold slip

We study the Stokes problems in a bounded planar domain Ω with a friction type boundary condition that switches between a slip and no-slip stage. Our main goal is to determine under which conditions concerning the smoothness of Ω solutions to the Stokes system with the slip boundary conditions depend continuously on variations of Ω. Having this result at our disposal, we easily prove the existence of a solution to optimal shape design problems for a large class of cost functionals. In order to release the impermeability condition, whose numerical treatment could be troublesome, we use a penalty approach. We introduce a family of shape optimization problems with the penalized state relations. Finally we establish convergence properties between solutions to the original and modified shape optimization problems when the penalty parameter tends to zero.     

ALMOST OPTIMAL LOCAL WELL-POSEDNESS OF THE MAXWELL-KLEIN-GORDON EQUATIONS IN 1 + 4 DIMENSIONS

Abstract

We study strong solutions of the Navier–Stokes equations for nonhomogeneous incompressible fluids in Ω ? R ³. Deriving higher a priori estimates independent of the lower bounds of the density, we prove the existence and uniqueness of local strong solutions to the initial value problem (for Ω =R ³) or the initial boundary value problem (for Ω ? ? R ³) even though the initial density vanishes in an open subset of Ω, i.e., an initial vacuum exists. As an immediate consequence of the a priori estimates, we obtain a continuation theorem for the local strong solutions.     

Stationary solutions to an energy model for semiconductor devices where the equations are defined on different domains

We discuss a stationary energy model from semiconductor modelling. We accept the more realistic assumption that the continuity equations for electrons and holes have to be considered only in a subdomain Ω₀ of the domain of definition Ω of the energy balance equation and of the Poisson equation. Here Ω₀ corresponds to the region of semiconducting material, Ω \ Ω₀ represents passive layers. Metals serving as contacts are modelled by Dirichlet boundary conditions. We prove a local existence and uniqueness result for the two‐dimensional stationary energy model. For this purpose we derive a W^1,p ‐regularity result for solutions of systems of elliptic equations with different regions of definition and use the Implicit Function Theorem. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Laplacian vector fields theory in domains with rectifiable boundary

Given a domain Ω in ?³ with rectifiable boundary, we consider main integral, and some other, theorems for the theory of Laplacian (sometimes called solenoidal and irrotational, or harmonic) vector fields paying a special attention to the problem of decomposing a continuous vector field, with an additional condition, u on the boundary Γof Ω ? ?³ into a sum u = u⁺+u^? were u^± are boundary values of vector fields which are Laplacian in Ω and its complement respectively. Our proofs are based on the intimate relations between Laplacian vector fields theory and quaternionic analysis for the Moisil–Theodorescu operator. Copyright © 2006 John Wiley & Sons, Ltd.     

Optimal error estimates of the local discontinuous Galerkin method for nonlinear second‐order elliptic problems on Cartesian grids

In this paper, we study the local discontinuous Galerkin (LDG) methods for two‐dimensional nonlinear second‐order elliptic problems of the type u_xx + u_yy = f(x, y, u, u_x, u_y) , in a rectangular region Ω with classical boundary conditions on the boundary of Ω . Convergence properties for the solution and for the auxiliary variable that approximates its gradient are established. More specifically, we use the duality argument to prove that the errors between the LDG solutions and the exact solutions in the L² norm achieve optimal (p + 1)th order convergence, when tensor product polynomials of degree at most p are used. Moreover, we prove that the gradient of the LDG solution is superclose with order p + 1 toward the gradient of Gauss–Radau projection of the exact solution. The results are valid in two space dimensions on Cartesian meshes using tensor product polynomials of degree p ≥ 1 , and for both mixed Dirichlet–Neumann and periodic boundary conditions. Preliminary numerical experiments indicate that our theoretical findings are optimal.     

Boundary concentration phenomena for a singularly perturbed elliptic problem

We exhibit new concentration phenomena for the equation ? ε² Δu + u = u^p in a smooth bounded domain Ω ? ?² and with Neumann boundary conditions. The exponent p is greater than or equal to 2 and the parameter ε is converging to 0. For a suitable sequence ε_n → 0 we prove the existence of positive solutions u_n concentrating at the whole boundary of Ω or at some component. © 2002 Wiley Periodicals, Inc.     

On the weak solution of the Neumann problem for the 2D Helmholtz equation in a convex cone and Hs regularity

We extend previous results for the Neumann boundary value problem to the case of boundary data from the space $H^{-\frac{1}{2}+\varepsilon}(\Gamma), 0{<}{\varepsilon}{<}\frac{1}{2}We extend previous results for the Neumann boundary value problem to the case of boundary data from the space $H^{-\frac{1}{2}+\varepsilon}(\Gamma), 0{<}{\varepsilon}{<}\frac{1}{2}$, where Γ=?Ω is the boundary of a two‐dimensional cone Ω with angle β<π. We prove that for these boundary conditions the solution of the Helmholtz equation in Ω exists in the Sobolev space H^1+ε(Ω), is unique and depends continuously on the boundary data. Moreover, we give the Sommerfeld representation for these solutions. This can be used to formulate explicit compatibility conditions on the data for regularity properties of the corresponding solution. Copyright © 2010 John Wiley & Sons, Ltd.     

ON A BI-HARMONIC EQUATION INVOLVING CRITICAL EXPONENT: EXISTENCE AND MULTIPLICITY RESULTS

In this paper, we consider the problem of existence as well as multiplicity results for a bi-harmonic equation under the Navier boundary conditions: △2 u = K(x)u p , u > 0 in Ω , △u = u = 0 on Ω , where Ω is a smooth domain in R n , n 5, and p + 1 = 2 n n 4 is the critical Sobolev exponent. We obtain highlightly a new criterion of existence, which provides existence results for a dense subset of positive functions, and generalizes Bahri-Coron type criterion in dimension six. Our argument gives also estimates on the Morse index of the obtained solutions and extends some known results. Moreover, it provides, for generic K, Morse inequalities at infinity, which delivers lower bounds for the number of solutions. As further applications of this Morse theoretical approach, we prove more existence results.     

Solvability of an initial boundary value problem for the euler equations in twodimensional domain with corners

The aim of this paper is to prove the existence and uniqueness of local solutions of some initial boundary value problems for the Euler equations of an incompressible fluid in a bounded domain Ω ? R ² with corners. We consider two cases of a nonvanishing normal component of velocity on the boundary. In three-dimensional case such problems have been considered in papers [12], [13], [14]. Similar problems in domains without corners have been considered in [2]–[6], [11]. In this paper the relation between the maximal corner angle of the boundary and the smoothness of the solutions is shown. The paper consists of four sections. In section 1 two initial boundary value problems for the Euler equations are formulated. In section 2 the existence and uniqueness of solutions of the Laplace equation in twodimensional domain with corners for the Dirichlet and Neumann problems is proved in the Sobolev spaces. In sections 3 and 4 we prove the existence and uniqueness of solutions of problems formulated in section 1, using the method of successive approximations.     

Mixed boundary value problems for nonlinear elliptic equations in multidimensional non-smooth domains

The nonlinear elliptic equation is investigated. It is supposed that u fulfils a mixed boundary value condition and that Ω ? IRⁿ (n ≥ 3) has a piecewise smooth boundary. W^s,2 — regularity (s < 3/2) of u and L^p — properties of the first and the second derivatives of u are proven.     

A nonexistence theorem for an equation with critical Sobolev exponent in the half space

In this paper we prove the nonexistence of positive solutions of the equation-Δu=u^2*-1 inR ₊ ^N with certain homogeneous mixed boundary conditions. The proof is based on a monotonicity theorem obtained using the moving plane methods and some recent results of Berestycki and Nirenberg (see [BN]). The nonexistence theorem is applied to improve a result of [GP] on the characterization of the critical levels of a functional related to some nonlinear elliptic problem with critical Sobolev exponent and mixed boundary conditions.     

An exterior boundary value problem in Minkowski space

In three‐dimensional Lorentz–Minkowski space ??³, we consider a spacelike plane Π and a round disc Ω over Π. In this article we seek the shapes of unbounded surfaces whose boundary is ? Ω and its mean curvature is a linear function of the distance to Π. These surfaces, called stationary surfaces, are solutions of a variational problem and governed by the Young–Laplace equation. In this sense, they generalize the surfaces with constant mean curvature in ??³. We shall describe all axially symmetric unbounded stationary surfaces with special attention in the case that the surface is asymptotic to Π at the infinity. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A symmetry-breaking problem in the annulus

We study questions of degeneracy and bifurcation for radial solutions of the semilinear elliptic equation ?u(x) + f(u(x)) = 0, x isin; [math001], [math001]an annulus in Rⁿ, with homogeneous Dirichlet boundary conditions. For certain nonlinearities f(u), we prove existence of degenerate radial solutions u (for which the kernel of the linearized operator Lz = ?z + [math001](u)z, z isin; C²We study questions of degeneracy and bifurcation for radial solutions of the semilinear elliptic equation ?u(x) + f(u(x)) = 0, x isin; [math001], [math001]an annulus in Rⁿ, with homogeneous Dirichlet boundary conditions. For certain nonlinearities f(u), we prove existence of degenerate radial solutions u (for which the kernel of the linearized operator Lz = ?z + [math001](u)z, z isin; C²$⁰([math001]), is non-trivial) and existence of nonradial solutions for the semi-linear equation. These nonradial (asymmetric) solutions are obtained via a bifurcation procedure from the radial (symmetric) ones. This phenomena is called symmetry-breaking. The bifurcation results are proved by a Conley index argument     

Asymptotic expansions of the pressure and the free boundary for flows through porous media

In this paper we discuss the free boundary value problem for the pressure head u of a compressible fluid flowing through a homogeneous porous medium. This process is governed by the partial differential equation ∈?_tu-? _x ² u=0, where ∈ is proportional to the compressibility of the fluid. We shall show that the pressure as well as the free boundary converge to the corresponding stationary solutions when ∈ tends to zero and shall furthermore estimate the error in terms of powers of ∈. Roughly speaking in the case of water, for example, this means that if we neglect its compressibility, which indeed is very small, we can estimate the error.