A conformal mapping algorithm for the Bernoulli free boundary value problem

We propose a new numerical method for the solution of the Bernoulli free boundary value problem for harmonic functions in a doubly connected domain D in where an unknown free boundary Γ₀ is determined by prescribed Cauchy data on Γ₀ in addition to a Dirichlet condition on the known boundary Γ₁. Our main idea is to involve the conformal mapping method as proposed and analyzed by Akduman, Haddar, and Kress for the solution of a related inverse boundary value problem. For this, we interpret the free boundary Γ₀ as the unknown boundary in the inverse problem to construct Γ₀ from the Dirichlet condition on Γ₀ and Cauchy data on the known boundary Γ₁. Our method for the Bernoulli problem iterates on the missing normal derivative on Γ₁ by alternating between the application of the conformal mapping method for the inverse problem and solving a mixed Dirichlet–Neumann boundary value problem in D. We present the mathematical foundations of our algorithm and prove a convergence result. Some numerical examples will serve as proof of concept of our approach. Copyright © 2015 John Wiley & Sons, Ltd.     

A nonlocal nonlinear diffusion equation in higher space dimensions

We study the initial-value problem for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation, in the whole RN, N?1, or in a bounded smooth domain with Neumann or Dirichlet boundary conditions. First, we prove the existence, uniqueness and the validity of a comparison principle for solutions of these problems. In RN we show that if initial data is bounded and compactly supported, then the solutions is compactly supported for all positive time t, this implies the existence of a free boundary. Concerning the Neumann problem, we prove that the asymptotic behavior of the solutions as t→∞, they converge to the mean value of the initial data. For the Dirichlet problem we prove that the asymptotic behavior of the solutions as t→∞, they converge to zero.     

Lösungen der poissongleichung und harmonische vektorfelder in unbeschränkten gebieten

In the first part of this paper we consider generalised solutions of the Poisson equation Δ U = F in open subsets of R _n(n ? 3) with Dirichlet or Neumann boundary data. We prove existence and uniqueness theorems, not only for the corresponding interior and exterior problems, but also for domains with boundaries extending to infinity. In the second part we discuss generalised harmonic fields in open subsets of R ₃ with vanishing Dirichlet or Neumann boundary condition.     

Boundary value problems for the Fitzhugh-Nagumo equations

This paper is concerned with the study of the FitzHugh-Nagumo equations. These equations arise in mathematical biology as a model of the transmission of electrical impulses through a nerve axon; they are a simplified version of the Hodgkin-Huxley equations. The FitzHugh-Nagumo equations consist of a non-linear diffusion equation coupled to an ordinary differential equation. v_t = v_xx + f(v) ? u, u_t = σv ? γu. We study these equations with either Dirichlet or Neumann boundary conditions, proving local and global existence, and uniqueness of the solutions. Furthermore, we obtain L_∞ estimates for the solutions in terms of the L₁ norm of the boundary data, when the boundary data vanish after a finite time and the initial data are zero. These estimates allow us to prove exponential decay of the solutions.     

On wellposedness in nonlinear acoustics

Motivated by a medical application from lithotripsy, we study the initial–boundary value problem given by Westervelt equation (1) in a bounded domain Ω. This models the nonlinear evolution of the acoustic pressure u excited at a part Γ₀ of the boundary. Along with the excitation given by Neumann boundary condition as in (1) , we also consider the Dirichlet type of excitation. Whereas shock waves are known to emerge after a sufficiently large time interval for appropriate initial and boundary conditions, we here prove existence and uniqueness as well as stability of a solution u for small data g, u₀ and u₁ or short time T, using a fixed point argument. Moreover we extend the result to the more general model given by the Kuznetsov equation (2) for the acoustic velocity potential ψ. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary layers for cellular flows at high Péclet numbers

We analyze the behavior of solutions of steady advection‐diffusion problems in bounded domains with prescribed Dirichlet data when the Péclet number Pe ? 1 is large. We show that the solution converges to a constant in each flow cell outside a boundary layer of width O(?^1/2), ? = Pe^?1, around the flow separatrices. We construct an ?‐dependent approximate “water pipe problem” purely inside the boundary layer that provides a good approximation of the solution of the full problem but has ?‐independent computational cost. We also define an asymptotic problem on the graph of streamline separatrices and show that solution of the water pipe problem itself may be approximated by an asymptotic, ?‐independent problem on this graph. Finally, we show that the Dirichlet‐to‐Neumann map of the water pipe problem approximates the Dirichlet‐to‐Neumann map of the separatrix problem with an error independent of the flow outside the boundary layers. © 2004 Wiley Periodicals, Inc.     

Classical solutions of two-dimensional stokes problems on non-smooth domains. Part 1: The radon integral operators

The applicability of the Neumann indirect method of potentials to the Dirichlet and Neumann problems for the two-dimensional Stokes operator on a non-smooth boundary Γ is subject to two kinds of sufficient and/or necessary conditions on Γ. The first one, occurring in electrostatic, is equivalent to the boundedness on C(Γ) of the velocity double-layer potential W as well as to the existence of jump relations of potentials. The second condition, which forces Γ to be a simple rectifiable curve and which, compared to the Laplacian, is a stronger restriction on the corners of Γ, states that the Fredholm radius of W is greater than 2. Under these conditions, the Radon boundary integral equations defined by the above-mentioned jump relations are solvable by the Fredholm theory; the double- (for Dirichlet) and the single- (for Neumann) layer potentials corresponding to their solutions are classical solutions of the Stokes problems.     

The lower bounds of life span of classical solutions to one‐dimensional initial‐Neumann boundary value problems for general quasilinear wave equations

In this paper, we will study the lower bounds of the life span (the maximal existence time) of solutions to the initial‐boundary value problems with small initial data and zero Neumann boundary data on exterior domain for one‐dimensional general quasilinear wave equations u_tt?u_xx=b(u,Du)u_xx+F(u,Du). Our lower bounds of the life span of solutions in the general case and special case are shorter than that of the initial‐Dirichlet boundary value problem for one‐dimensional general quasilinear wave equations. We clarify that although the lower bounds in this paper are same as that in the case of Robin boundary conditions obtained in the earlier paper, however, the results in this paper are not the trivial generalization of that in the case of Robin boundary conditions because the fundamental Lemmas 2.4, 2.5, 2.6, and 2.7, that is, the priori estimates of solutions to initial‐boundary value problems with Neumann boundary conditions, are established differently, and then the specific estimates in this paper are different from that in the case of Robin boundary conditions. Another motivation for the author to write this paper is to show that the well‐posedness of problem 1.1 is the essential precondition of studying the lower bounds of life span of classical solutions to initial‐boundary value problems for general quasilinear wave equations. The lower bound estimates of life span of classical solutions to initial‐boundary value problems is consistent with the actual physical meaning. Finally, we obtain the sharpness on the lower bound of the life span 1.8 in the general case and 1.10 in the special case. Copyright © 2015 John Wiley & Sons, Ltd.     

Degenerate triply nonlinear problems with nonhomogeneous boundary conditions

The paper addresses the existence and uniqueness of entropy solutions for the degenerate triply nonlinear problem: b(v) _t − div α(v, ▽g(v)) = f on Q:= (0, T) × Ω with the initial condition b(v(0, ·)) = b(v ₀) on Ω and the nonhomogeneous boundary condition “v = u” on some part of the boundary (0, T) × ∂Ω”. The function g is continuous locally Lipschitz continuous and has a flat region [A ₁, A ₂,] with A ₁ ≤ 0 ≤ A ₂ so that the problem is of parabolic-hyperbolic type.     

Identification of a source factor in a control problem for the heat equation with a boundary memory term

We consider a material with thermal memory occupying a bounded region Ω with boundary Γ. The evolution of the temperature u(t,x) is described by an integrodifferential parabolic equation containing a heat source of the form f(t)z₀(x). We formulate an initial and boundary value control problem based on a feedback device located on Γ and prescribed by means of a quite general memory operator. Assuming both u and the source factor f are unknown, we study the corresponding inverse and control problem on account of an additional information. We prove a result of existence and uniqueness of the solution (u,f). Copyright © 2015 John Wiley & Sons, Ltd.     

Dirichlet‐to‐Neumann Map for a Nonlinear Diffusion Equation

A nonlinear diffusive equation with moving boundaries is analyzed by constructing the corresponding Dirichlet‐to‐Neumann map. In particular, the Dirichlet boundary value and the initial condition are used to derive the unknown Neumann boundary value. Then, a contraction‐mapping technique is used to prove existence and uniqueness of the solution for small times.     

Finite energy solutions of self‐adjoint elliptic mixed boundary value problems

This paper describes existence, uniqueness and special eigenfunction representations of H¹‐solutions of second order, self‐adjoint, elliptic equations with both interior and boundary source terms. The equations are posed on bounded regions with Dirichlet conditions on part of the boundary and Neumann conditions on the complement. The system is decomposed into separate problems defined on orthogonal subspaces of H¹(Ω). One problem involves the equation with the interior source term and the Neumann data. The other problem just involves the homogeneous equation with Dirichlet data. Spectral representations of the solution operators for each of these problems are found. The solutions are described using bases that are, respectively, eigenfunctions of the differential operator with mixed null boundary conditions, and certain mixed Steklov eigenfunctions. These series converge strongly in H¹(Ω). Necessary and sufficient conditions for the Dirichlet part of the boundary data to have a finite energy extension are described. The solutions for a problem that models a cylindrical capacitor is found explicitly. Copyright © 2009 John Wiley & Sons, Ltd.     

On the existence of smooth self‐similar blowup profiles for the wave map equation

Consider the equivariant wave map equation from Minkowski space to a rotationally symmetric manifold N that has an equator (e.g., the sphere). In dimension 3, this paper presents a necessary and sufficient condition on N for the existence of a smooth self‐similar blowup profile. More generally, we study the relation between

• the minimizing properties of the equator map for the Dirichlet energy corresponding to the (elliptic) harmonic map problem and
• the existence of a smooth blowup profile for the (hyperbolic) wave map problem.

This has several applications to questions of regularity and uniqueness for the wave map equation. © 2008 Wiley Periodicals, Inc.     

On the integral equation method for the plane mixed boundary value problem of the Laplacian

Although the plane boundary value problem for the Laplacian with given Dirichlet data on one part Γ₂ and given Neumann data on the remaining part Γ₂ of the boundary is the simplest case of mixed boundary value problems, we present several applications in classical mathematical physics. Using Green's formula the problem is converted into a system of Fredholm integral equations for the yet unknown values of the solution u on Γ₂ and the also desired values of the normal derivatie on Γ₁. One of these equations has principal part of the second kind, whereas that one of the other is of the first kind. Since any improvement of constructive methods requires higher regularity of u but, on the other hand, grad u possesses singularities at the collision points Γ₁ ∩ Γ₂ even for C^∞ data, u is decomposed into special singular terms and a regular rest. This is incorporated into the integral equations and the modified system is solved in appropriate Sobolev spaces. The solution of the system requires to solve a Fredholm equation of the first kind on the arc Γ₂ providing an improvement of regularity for the smooth part of u. Since the integral equations form a strongly elliptic system of pseudodifferential operators, the Galerkin procedure converges. Using regular finite element functions on Γ₁ and Γ₂ augmented by the special singular functions we obtain optimal order of asymptotic convergence in the norm corresponding to the energy norm of u and also superconvergence as well as high orders in smoother norms if the given data are smooth (and not the solution).     

On a nonlocal generalization of the biharmonic Dirichlet problem

We consider a mixed problem with the Dirichlet boundary conditions and integral conditions for the biharmonic equation. We prove the existence and uniqueness of a generalized solution in the weighted Sobolev space W ₂². We show that the problem can be viewed as a generalization of the Dirichlet problem.     

The evolution dam problem for a compressible fluid with nonlinear Darcy's law and Dirichlet boundary condition

In this paper, we consider the evolution dam problem (P) related to a compressible fluid flow governed by a generalized nonlinear Darcy's law with Dirichlet boundary conditions on some part of the boundary. We establish existence of a solution for this problem. We choose a convenient regularized problem (P_?) for which we prove the existence and uniqueness of solution using the comparison Lemma 2.1 and the Schauder fixed‐point theorem. Then, we pass to the limit, when ? goes to 0, to get a solution for our problem. Moreover, we will see another approach for the incompressible case where we pass to the limit in (P), when α goes to 0, to get a solution.     

The Hele-Shaw problem with surface tension in a half-plane

In this paper, we consider the Hele-Shaw problem in a 2-dimensional fluid domain Ω(t) which is constrained to a half-plane. The boundary of Ω(t) consist of two components: Γ₀(t) which lies on the boundary of the half-plane, and Γ(t) which lies inside the half-plane. On Γ(t) we impose the classical boundary conditions with surface tension, and on Γ₀(t) we prescribe the normal derivative of the fluid pressure. At the point where Γ₀(t) and Γ(t) meet, there is an abrupt change in the boundary condition giving rise to a singularity in the fluid pressure. We prove that the problem has a unique solution with smooth free boundary Γ(t) for some small time interval.     

The Neumann problem for some degenerate elliptic equations

In the paper we study the equation L _u = f, where L is a degenerate elliptic operator, with Neumann boundary condition in a bounded open set μ. We prove existence and uniqueness of solutions in the space H(μ) for the Neumann problem.     

Some degenerate and quasilinear parabolic systems not in divergence form

This paper deals with positive solutions of degenerate and quasilinear parabolic systems not in divergence form: u_t=u^p(Δu+av), v_t=v^q(Δv+bu), with null Dirichlet boundary conditions and positive initial conditions, where p, q, a and b are all positive constants. The local existence and uniqueness of classical solution are proved. Moreover, it will be proved that all solutions exist globally if and only if ab?λ₁², where λ₁ is the first eigenvalue of −Δ in Ω with homogeneous Dirichlet boundary condition.