Regularity of flat free boundaries in two-phase problems for the p-Laplace operator

In this paper we continue the study in Lewis and Nyström (2010) [19], concerning the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator, by proving regularity of the free boundary assuming that the free boundary is close to a Lipschitz graph.     

Regularity of Lipschitz free boundaries in two-phase problems for the p-Laplace operator

In this paper we study the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator and we prove, in particular, that Lipschitz free boundaries are C1,γ-smooth for some γ∈(0,1). As part of our argument, and which is of independent interest, we establish a Hopf boundary type principle for non-negative p-harmonic functions vanishing on a portion of the boundary of a Lipschitz domain.     

Existence and regularity of the solution of a mixed boundary value problem for the Keldysh equation with a nonlinear absorption term

The Keldysh equation is a more general form of the classic Tricomi equation from fluid dynamics. Its well-posedness and the regularity of its solution are interesting and important. The Keldysh equation is elliptic in y>0 and is degenerate at the line y=0 in R². Adding a special nonlinear absorption term, we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain—similar to the potential fluid shock reflection problem. By means of an elliptic regularization technique, a delicate a priori estimate and compact argument, we show that the solution of a mixed boundary value problem of the Keldysh equation is smooth in the interior and Lipschitz continuous up to the degenerate boundary under some conditions. We believe that this kind of regularity result for the solution will be rather useful.     

Existence and regularity for a nonlinear boundary flow problem of population genetics

In this paper we consider a heat equation with nonlinear boundary condition occurring in population genetics, the selection–migration problem for alleles in a region, considering flow of genes throughout the boundary. Such a problem determines a gradient flow in a convenient phase space and then the dynamics for large times depends heavily on the knowledge of the equilibrium solutions. We address the questions of the existence of a nontrivial equilibrium solution and its regularity.     

Linear and nonlinear abstract equations with parameters

This paper focuses on nonlocal boundary value problems for linear and nonlinear abstract elliptic equations in Banach spaces. Here equations and boundary conditions contain certain parameters. The uniform separability of the linear problem and the existence and uniqueness of maximal regular solution of nonlinear problem are obtained in L_p spaces. For linear case the discreteness of spectrum of corresponding parameter dependent differential operator is obtained. The behavior of solution when the parameter approaches zero and its smoothness with respect to the parameter is established. Moreover, we show the estimate for analytic semigroups in terms of interpolation spaces. This fact can be used to obtain maximal regularity properties for abstract boundary value problems.     

Application of the G class of functions in the parabolic class

In this paper,the application of the G class of functions in the parabolic class is considered. The regularity of the solution for the first boundary value problem of parabolic equation in divergence form is proved.     

Solution of the Dirichlet and Neumann problems for a modified Helmholtz equation in Besov spaces on an annulus

Here we study Dirichlet and Neumann problems for a special Helmholtz equation on an annulus. Our main aim is to measure smoothness of solutions for the boundary datum in Besov spaces. We shall use operator theory to solve this problem. The most important advantage of this technique is that it enables to consider equations in vector-valued settings. It is interesting to note that optimal regularity of this problem will be a special case of our main result.     

Boundary regularity for nonlinear elliptic systems

We consider questions of boundary regularity for solutions of certain systems of second-order nonlinear elliptic equations. We obtain a general criterion for a weak solution to be regular in the neighbourhood of a given boundary point. The proof yields directly the optimal regularity for the solution in this neighbourhood. This result is new for the situation under consideration (general nonlinear second order systems in divergence form, with inhomogeneity obeying the natural growth conditions). Received: 6 July 2001 / Accepted: 27 September 2001 / Published online: 28 February 2002     

Higher regularity for free boundary and tangential derivative problems and global solutions to regular reflections for the unsteady transonic small disturbance (UTSD) equations

We establish C^2,α$C^{2,\alpha }$-estimates for solutions of a class of quasilinear elliptic equations with free boundary and tangential derivative boundary problems. Using this regularity result we show the existence of global solutions to regular shock reflections for the unsteady transonic small disturbance (UTSD) equation. We also present Lipschitz estimates near the degenerate Dirichlet boundary (the sonic boundary) for the UTSD equation.     

Existence and uniqueness results for large solutions of general nonlinear elliptic equations

We study under what condition there exists a solution of -u+f(u)=0 in a domain which blows-up on the boundary, independently of the regularity of the boundary, and we provide criteria for uniqueness. We apply our results to the case f(u)= e^au.     

Weak solutions for nonlocal boundary value problems with low regularity data

This paper concerns the existence and uniqueness of weak solutions for elliptic and parabolic equations under nonlocal boundary conditions, based on maximal regularity. It also gives the positivity of solutions which can be used in monotone iteration methods. As an application, the results are used to discuss some specific nonlocal problems.     

Fine regularity for elliptic and parabolic anisotropic Robin problems with variable exponents

We investigate a class of quasi-linear elliptic and parabolic anisotropic problems with variable exponents over a general class of bounded non-smooth domains, which may include non-Lipschitz domains, such as domains with fractal boundary and rough domains. We obtain solvability and global regularity results for both the elliptic and parabolic Robin problem. Some a priori estimates, as well as fine properties for the corresponding nonlinear semigroups, are established. As a consequence, we generalize the global regularity theory for the Robin problem over non-smooth domains by extending it for the first time to the variable exponent case, and furthermore, to the anisotropic variable exponent case.     

General non local boundary value problem for second order elliptic equation

In this paper we study a class of second order coefficient operators differential equation with general (possibly non local) boundary conditions. We obtain new results extending those given in a previous paper 1 . Existence, uniqueness and optimal regularity of the strict solution are proved in UMD spaces, using the well‐known Dore–Venni theorem.     

Energy minimizing harmonic maps with an obstacle at the free boundary

We consider partial regularity for energy minimizing maps satisfying a partially free boundary condition. This condition takes the form of the requirement that a relatively open subset of the boundary of the domain manifold be mapped into a closed submanifold with non-empty boundary, contained in the target manifold. We obtain an optimal estimate on the Hausdorff dimension of the singular set of such a map. Our result can be interpreted as regularity result for a vector-valued Signorini, or thin-obstacle, problem.     

Monotonicity theorems for Laplace Beltrami operator on Riemannian manifolds

For free boundary problems on Euclidean spaces, the monotonicity formulas of Alt–Caffarelli–Friedman and Caffarelli–Jerison–Kenig are cornerstones for the regularity theory as well as the existence theory. In this article we establish the analogs of these results for the Laplace–Beltrami operator on Riemannian manifolds. As an application we show that our monotonicity theorems can be employed to prove the Lipschitz continuity for the solutions of a general class of two-phase free boundary problems on Riemannian manifolds.     

On the two-phase membrane problem with coefficients below the Lipschitz threshold

We study the regularity of the two-phase membrane problem, with coefficients below the Lipschitz threshold. For the Lipschitz coefficient case one can apply a monotonicity formula to prove the C1,1$C^{1,1}$-regularity of the solution and that the free boundary is, near the so-called branching points, the union of two C¹$C^1$-graphs. In our case, the same monotonicity formula does not apply in the same way. In the absence of a monotonicity formula, we use a specific scaling argument combined with the classification of certain global solutions to obtain C1,1$C^{1,1}$-estimates. Then we exploit some stability properties with respect to the coefficients to prove that the free boundary is the union of two Reifenberg vanishing sets near so-called branching points.     

Existence and regularity of multiple solutions for infinitely degenerate nonlinear elliptic equations with singular potential

In this paper, we study the Dirichlet problem for a class of infinitely degenerate nonlinear elliptic equations with singular potential term. By using the logarithmic Sobolev inequality and Hardy's inequality, the existence and regularity of multiple nontrivial solutions have been proved.     

Superlinear elliptic equations with singular coefficients on the boundary

In this paper, a superlinear elliptic equation whose coefficient diverges on the boundary is studied in any bounded domain Ω under the zero Dirichlet boundary condition. Although the equation has a singularity on the boundary, a solution is smooth on the closure of the domain. Indeed, it is proved that the problem has a positive solution and infinitely many solutions without positivity, which belong to or . Moreover, it is proved that a positive solution has a higher order regularity up to .     

Nonlinear degenerate elliptic equations and axially symmetric problems

We consider semilinear elliptic equations with a principal part degenerating on a boundary hyperplane. Weak existence, uniqueness and regularity of solutions are established by variational methods and by reduction to uniformly elliptic equations. An important application arises in the mathematical treatment of the rotating star problem in general relativity, where the axial symmetry admits the reduction of one of the Einstein equations to a problem of the above form on a meridian half plane. Received February 12, 1997 / Accepted May 15, 1997     

A minimum problem with free boundary in Orlicz spaces

We consider the optimization problem of minimizing in the class of functions W1,G(Ω) with , for a given φ₀?0 and bounded. W1,G(Ω) is the class of weakly differentiable functions with . The conditions on the function G allow for a different behavior at 0 and at ∞. We prove that every solution u is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, Ω∩∂{u>0}, is a regular surface. Also, we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the Lipschitz regularity of the weak solutions and the C1,α regularity of their free boundaries near “flat” free boundary points.