Regularity of the free boundary in two-phase problems for linear elliptic operators

In this paper we complete the study of the regularity of the free boundary in two-phase problems for linear elliptic operators started in [M.C. Cerutti, F. Ferrari, S. Salsa, Two-phase problems for linear elliptic operators with variable coefficients: Lipschitz free boundaries are C1,γ, Arch. Ration. Mech. Anal. 171 (2004) 329-348]. In particular we prove that Lipschitz and flat free boundaries (in a suitable sense) are smooth. As byproduct, we prove that Lipschitz free boundaries are smooth in the case of quasilinear operators of the form div(A(x,u)∇u) with Lipschitz coefficients.     

Anisotropic elliptic equations with VMO coefficients

This paper presents the study of maximal regularity properties for anisotropic differential-operator equations with VMO (vanishing mean oscillation) coefficients. We prove that the corresponding differential operator is separable and is a generator of analytic semigroup in vector-valued L^p spaces. Moreover, discreetness of spectrum and completeness of root elements of this operator is obtained.     

Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations

In this paper, we study the local behavior of the solutions to the three-dimensional magnetohydrodynamic equations. we are interested in both the uniform gradient estimates for smooth solutions and regularity of weak solutions. It is shown that, in some neighborhood of (x₀,t₀), the gradients of the velocity field u and the magnetic field B are locally uniformly bounded in L^∞ norm as long as that either the scaled local L²-norm of the gradient or the scaled local total energy of the velocity field is small, and the scaled local total energy of the magnetic field is uniformly bounded. These estimates indicate that the velocity field plays a more dominant role than that of the magnetic field in the regularity theory. As an immediately corollary we can derive an estimates of Hausdorff dimension on the possible singular set of a suitable weak solution as in the case of pure fluid. Various partial regularity results are obtained as consequences of our blow-up estimates.     

Existence and Regularity of Solutions to Model for Liquid Mixture of He-He

Existence and regularity of solutions to model for liquid mixture of ³He-⁴He is considered in this paper. First, it is proved that this system possesses a unique global weak solution in H¹(ω,C×R) by using Galerkin method. Secondly, by using an iteration procedure, regularity estimates for the linear semigroups, it is proved that the model for liquid mixture of ³He-⁴He has a unique solution in H^k(ω,C×R) for all k ≥ 1.     

Regularity of variational solutions to linear boundary value problems in Lipschitz domains

In a bounded Lipschitz domain in ?ⁿ, we consider a second-order strongly elliptic system with symmetric principal part written in divergent form. We study the Neumann boundary value problem in a generalized variational (or weak) setting using the Lebesgue spaces H _p ^σ (Ω) for solutions, where p can differ from 2 and σ can differ from 1. Using the tools of interpolation theory, we generalize the known theorem on the regularity of solutions, in which p = 2 and {σ ? 1} < 1/2, and the corresponding theorem on the unique solvability of the problem (Savaré, 1998) to p close to 2. We compare this approach with the nonvariational approach accepted in numerous papers of the modern theory of boundary value problems in Lipschitz domains. We discuss the regularity of eigenfunctions of the Dirichlet, Neumann, and Poincaré-Steklov spectral problems.     

On a nonlinear elliptic-parabolic integro-differential equation with L-data

We generalize the concept of entropy solutions for parabolic equations with L¹-data and consider a class of nonlinear history-dependent degenerated elliptic-parabolic equations including problems with a fractional time derivative such as with Dirichlet boundary conditions and initial condition, where 0<γ?1. We show uniqueness of entropy solutions for general L¹-data by Kruzhkov's method of doubling variables. Moreover, existence in the nondegenerated case, i.e. b≡id, is shown by using the concept of generalized solutions of a corresponding abstract Volterra equation.     

Concentration-compactness phenomena in the higher order Liouville's equation

We investigate different concentration-compactness and blow-up phenomena related to the Q-curvature in arbitrary even dimension. We first treat the case of an open domain in R2m, then that of a closed manifold and, finally, the particular case of the sphere S2m. In all cases we allow the sign of the Q-curvature to vary, and show that in the case of a closed manifold, contrary to the case of open domains in R2m, blow-up phenomena can occur only at points of positive Q-curvature. As a consequence, on a locally conformally flat manifold of non-positive Euler characteristic we always have compactness.     

Initial boundary value problems for nonlinear dispersive wave equations

In this paper we study initial value boundary problems of two types of nonlinear dispersive wave equations on the half-line and on a finite interval subject to homogeneous Dirichlet boundary conditions. We first prove local well-posedness of the rod equation and of the b-equation for general initial data. Furthermore, we are able to specify conditions on the initial data which on the one hand guarantee global existence and on the other hand produce solutions with a finite life span. In the case of finite time singularities we are able to describe the precise blow-up scenario of breaking waves. Our approach is based on sharp extension results for functions on the half-line or on a finite interval and several symmetry preserving properties of the equations under discussion.     

Asymptotic regularity and attractors of the reaction-diffusion equation with nonlinear boundary condition

We consider the dynamical behavior of the reaction-diffusion equation with nonlinear boundary condition for both autonomous and non-autonomous cases. For the autonomous case, under the assumption that the internal nonlinear term f is dissipative and the boundary nonlinear term g is non-dissipative, the asymptotic regularity of solutions is proved. For the non-autonomous case, we obtain the existence of a compact uniform attractor in H¹(Ω) with dissipative internal and boundary nonlinearities.     

Pointwise Estimates for Laplace Equation. Applications to the Free Boundary of the Obstacle Problem with Dini Coefficients

In this paper we are interested in pointwise regularity of solutions to elliptic equations. In a first result, we prove that if the modulus of mean oscillation of Δu at the origin is Dini (in L ^p average), then the origin is a Lebesgue point of continuity (still in L ^p average) for the second derivatives D ² u. We extend this pointwise regularity result to the obstacle problem for the Laplace equation with Dini right hand side at the origin. Under these assumptions, we prove that the solution to the obstacle problem has a Taylor expansion up to the order 2 (in the L ^p average). Moreover we get a quantitative estimate of the error in this Taylor expansion for regular points of the free boundary. In the case where the right hand side is moreover double Dini at the origin, we also get a quantitative estimate of the error for singular points of the free boundary. Our method of proof is based on some decay estimates obtained by contradiction, using blow-up arguments and Liouville Theorems. In the case of singular points, our method uses moreover a refined monotonicity formula.      

Elliptic boundary value problems with λ-dependent boundary conditions

In this paper second order elliptic boundary value problems on bounded domains Ω⊂Rn with boundary conditions on ∂Ω depending nonlinearly on the spectral parameter are investigated in an operator theoretic framework. For a general class of locally meromorphic functions in the boundary condition a solution operator of the boundary value problem is constructed with the help of a linearization procedure. In the special case of rational Nevanlinna or Riesz-Herglotz functions on the boundary the solution operator is obtained in an explicit form in the product Hilbert space L²(Ω)⊕(L²m(∂Ω)), which is a natural generalization of known results on λ-linear elliptic boundary value problems and λ-rational boundary value problems for ordinary second order differential equations.     

C1,1 regularity for degenerate complex Monge–Ampère equations and geodesic rays

We prove a C^1,1 estimate for solutions of complex Monge–Ampère equations on compact Kähler manifolds with possibly nonempty boundary, in a degenerate cohomology class. This strengthens previous estimates of Phong–Sturm. As applications we deduce the local C^1,1 regularity of geodesic rays in the space of Kähler metrics associated to a test configuration, as well as the local C^1,1 regularity of quasi-psh envelopes in nef and big classes away from the non-Kähler locus.     

Regularity criteria for suitable weak solutions of the Navier-Stokes equations near the boundary

We present some new regularity criteria for “suitable weak solutions” of the Navier-Stokes equations near the boundary in dimension three. We prove that suitable weak solutions are Hölder continuous up to the boundary provided that the scaled mixed norm with 3/p+2/q?2, 2<q?∞, (p,q)≠(3/2,∞) is small near the boundary. Our methods yield new results in the interior case as well. Partial regularity of weak solutions is also analyzed under some additional integral conditions.     

Blow-up in a class of non-linear parabolic problems with time-dependent coefficients under Robin type boundary conditions

The goal of this article is the determination of upper and lower bounds for the blow-up time t ^☆ for a class of non-linear parabolic problems with time dependent coefficients under Robin type boundary conditions.     

Some properties and characterizations for generalized multivariate Pareto distributions

In this paper, several distributional properties and characterization theorems of the generalized multivariate Pareto distributions are studied. It is found that the multivariate Pareto distributions have many mixture properties. They are mixed either by geometric, Weibull, or exponential variables. The multivariate Pareto, MP^(k)(I), MP^(k)(II), and MP^(k)(IV) families have closure property under finite sample minima. The MP^(k)(III) family is closed under both geometric minima and geometric maxima. Through the geometric minima procedure, one characterization theorem for MP^(k)(III) distribution is developed. Moreover, the MP^(k)(III) distribution is proved as the limit multivariate distribution under repeated geometric minimization. Also, a characterization theorem for the homogeneous MP^(k)(IV) distribution via the weighted minima among the ordered coordinates is developed. Finally, the MP^(k)(II) family is shown to have the truncation invariant property.     

The regularity of the surface of the Gauss curvature 0≤K ∈ C 0 ∞

Under the mild conditions, it is proved that the convex surface is global C^1.1, with the given Gaussian curvature 0≤K ∈ C ₀ ^∞ and the given boundary curve. Examples are given to show that the regularity is optimal. Project supported by the Doctoral Funds of China and the National Natural Science Foundation of China (Grant No. 19771009).     

Multiple positive solutions for multipoint boundary value problems with one-dimensional p-Laplacian

In this paper we consider the multiplicity of positive solutions for the one-dimensional p-Laplacian differential equation (?p^′(u^′))+q(t)f(t,u,u^′)=0, t∈(0,1), subject to some boundary conditions. By means of a fixed point theorem due to Avery and Peterson, we provide sufficient conditions for the existence of multiple positive solutions to some multipoint boundary value problems.     

A trace regularity result for thermoelastic equations with application to optimal boundary control

We consider a mixed problem for a Kirchoff thermoelastic plate model with clamped boundary conditions. We establish a sharp regularity result for the outer normal derivative of the thermal velocity on the boundary. The proof, based upon interpolation techniques, benefits from the exceptional regularity of traces of solutions to the elastic Kirchoff equation. This result, which complements recent results obtained by the second and third authors, is critical in the study of optimal control problems associated with the thermoelastic system when subject to thermal boundary control. Indeed, the present regularity estimate can be interpreted as a suitable control-theoretic property of the corresponding abstract dynamics, which is crucial to guarantee well-posedness for the associated differential Riccati equations.     

On the concentration properties for the nonlinear Schrödinger equation with a Stark potential

In this paper, we study blow-up solutions of the Cauchy problem to the L2 critical nonlinear Schrdinger equation with a Stark potential. Using the variational characterization of the ground state for nonlinear Schrdinger equation without any potential, we obtain some concentration properties of blow-up solutions, including that the origin is the blow-up point of the radial blow-up solutions, the phenomenon of L2-concentration and rate of L2-concentration of blow-up solutions.     

Multiple positive solutions for boundary value problems of second order delay differential equations with one-dimensional p-Laplacian

We consider the boundary value problems: (?p(x^′′(t)))+q(t)f(t,x(t),x(t−1),x^′(t))=0, ?p(s)=|s|^p−2s, p>1, t∈(0,1), subject to some boundary conditions. By using a generalization of the Leggett-Williams fixed-point theorem due to Avery and Peterson, we provide sufficient conditions for the existence of at least three positive solutions to the above problems.