Bernoulli Free Boundary Problem Research Articles

A weakly coupled system of p-Laplace type in a heat conduction problem

Abstract Given is a bounded domain Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} , and a vector-valued function defined on ∂ ⁡ Ω {\partial\Omega} (depicting temperature distributions from different sources), our objective is to study the mathematical model of a physical problem of enclosing ∂ ⁡ Ω {\partial\Omega} with a specific volume of insulating material to reduce heat loss in a stationary scenario. Mathematically, this task involves identifying a vector-valued function 𝐮 = ( u 1 , … , u m ) {\mathbf{u}=(u^{1},\dots,u^{m})} ( m ≥ 1 {m\geq 1} ) that represents the temperature within Ω and gives rise to a free boundary, somehow reminiscent of, but not equivalent to, the Bernoulli free boundary problem.

Regularity for one-phase Bernoulli problems with discontinuous weights and applications

We study a one-phase Bernoulli free boundary problem with weight function admitting a discontinuity along a smooth jump interface. In any dimension N ≥ 2 N\ge 2 , we show the C 1 , α C^{1, \alpha } regularity of the free boundary outside of a singular set of Hausdorff dimension at most N − 3 N-3 . In particular, we prove that the free boundaries are C 1 , α C^{1, \alpha } regular in dimension N = 2 N=2 , while in dimension N = 3 N=3 the singular set can contain at most a finite number of points. We use this result to construct singular free boundaries in dimension N = 2 N=2 , which are minimizing for one-phase functionals with weight functions in L ∞ L^\infty that are arbitrarily close to a positive constant.

On convex comparison for exterior Bernoulli problems with discontinuous anisotropy

We give a new, intuitive proof of a convex comparison principle for exterior Bernoulli free boundary problems with discontinuous anisotropy.

(Quasi-)conformal methods in two-dimensional free boundary problems

In this paper, we relate the theory of quasi-conformal maps to the regularity of the solutions to nonlinear thin-obstacle problems; we prove that the contact set is locally a finite union of intervals and apply this result to the solutions of one-phase Bernoulli free boundary problems with geometric constraint. We also introduce a new conformal hodograph transform, which allows to obtain the precise expansion at branch points of both the solutions to the one-phase problem with geometric constraint and a class of symmetric solutions to the two-phase problem, as well as to construct examples of free boundaries with cusp-like singularities.

On the interior Bernoulli free boundary problem for the fractional Laplacian on an interval

AbstractWe study the structure of solutions of the interior Bernoulli free boundary problem for $$(-\Delta )^{\alpha /2}$$ ( - Δ ) α / 2 on an interval D with parameter $$\lambda > 0$$ λ > 0 . In particular, we show that there exists a constant $$\lambda _{\alpha ,D} > 0$$ λ α , D > 0 (called the Bernoulli constant) such that the problem has no solution for $$\lambda \in (0,\lambda _{\alpha ,D})$$ λ ∈ ( 0 , λ α , D ) , at least one solution for $$\lambda = \lambda _{\alpha ,D}$$ λ = λ α , D and at least two solutions for $$\lambda > \lambda _{\alpha ,D}$$ λ > λ α , D . We also study the interior Bernoulli problem for the fractional Laplacian for an interval with one free boundary point. We discuss the connection of the Bernoulli problem with the corresponding variational problem and present some conjectures. In particular, we show for $$\alpha = 1$$ α = 1 that there exists solutions of the interior Bernoulli free boundary problem for $$(-\Delta )^{\alpha /2}$$ ( - Δ ) α / 2 on an interval which are not minimizers of the corresponding variational problem.

A physics-informed learning approach to Bernoulli-type free boundary problems

• A PINN methodology for a Bernoulli free boundary variational problem is proposed. • Quasi Monte-Carlo integration methods are used in the learning process of the PINN. • A redistribution strategy of domain points based on solution's Laplacian is developed. • Analysis on convex and non-convex domains assess the PINN reliability. Bernoulli free boundary problems (BFBP) govern many real applications, from chemistry to fluid dynamics. BFBP are overdetermined differential models in which the boundary of the solution's domain appears as an unknown. This work provides a variational formulation for BFBP, and a computational method focused on the novel methodology of Physics Informed Neural Networks (PINNs) is proposed. It consists in training a neural network to approximate the solution of the differential problem, minimizing a suitable cost function built, taking into account the physics constraint given by the model. In particular, since physics laws are injected through a loss function containing integral terms in our case, an approach based on quasi Monte-Carlo integration methods has been implemented. Moreover, an adaptive strategy to increase the model accuracy, exploiting topological information related to the shape of the solution, has been developed. Finally, when available, comparisons with the analytical solutions and studies in the case of non-convex domains assess the reliability of the PINN approach in the examined context.

On the Bernoulli free boundary problems for the half Laplacian and for the spectral half Laplacian

We study the exterior and interior Bernoulli problems for the half Laplacian and the interior Bernoulli problem for the spectral half Laplacian. We concentrate on the existence and geometric properties of solutions. Our main results are the following. For the exterior Bernoulli problem for the half Laplacian, we show that under starshapedness assumptions on the data the free domain is starshaped. For the interior Bernoulli problem for the spectral half Laplacian, we show that under convexity assumptions on the data the free domain is convex and we prove a Brunn–Minkowski inequality for the Bernoulli constant. For Bernoulli problems for the half Laplacian we use a variational methods in combination with regularity results for viscosity solutions, whereas for Bernoulli problem for the spectral half Laplacian we use the Beurling method based on subsolutions.

Interface regularity for semilinear one-phase problems

We study critical points of a one-parameter family of functionals arising in combustion models. The problems we consider converge, for infinitesimal values of the parameter, to Bernoulli's free boundary problem, also known as one-phase problem. We prove a C1,α estimates for the “interfaces” (level sets separating the burnt and unburnt regions). As a byproduct, we obtain the one-dimensional symmetry of minimizers in the whole RN, for N≤4, answering positively a conjecture of Fernández-Real and Ros-Oton.Our results are to Bernoulli's free boundary problem what Savin's results for the Allen-Cahn equation are to minimal surfaces.

The simultaneous asymmetric perturbation method for overdetermined free boundary problems

In this paper, we introduce a new method for applying the implicit function theorem to find nontrivial solutions to overdetermined problems with a fixed boundary (given) and a free boundary (to be determined). The novelty of this method lies in the kind of perturbations considered. Indeed, we work with perturbations that exhibit different levels of regularity on each boundary. This allows us to construct solutions (whose given boundary and free boundary exhibit different regularities) that would have been out of reach via more simple perturbation techniques. Another benefit of this method lies in the improvement of the regularity gap that we get between the free boundary and the boundary of the given domain (this can be interpreted as a “smoothing effect”). Moreover, we show how to employ this method to construct solutions to both the Bernoulli free boundary problem and the two-phase Serrin’s overdetermined problem near radially symmetric configurations. Finally, some geometric properties of the solutions, such as symmetry and convexity, are also discussed.

Comparison of Shape Derivatives Using CutFEM for Ill-posed Bernoulli Free Boundary Problem

In this paper we study and compare three types of shape derivatives for free boundary identification problems. The problem takes the form of a severely ill-posed Bernoulli problem where only the Dirichlet condition is given on the free (unknown) boundary, whereas both Dirichlet and Neumann conditions are available on the fixed (known) boundary. Our framework resembles the classical shape optimization method in which a shape dependent cost functional is minimized among the set of admissible domains. The position of the domain is defined implicitly by the level set function. The steepest descent method, based on the shape derivative, is applied for the level set evolution. For the numerical computation of the gradient, we apply the Cut Finite Element Method (CutFEM), that circumvents meshing and re-meshing, without loss of accuracy in the approximations of the involving partial differential models. We consider three different shape derivatives. The first one is the classical shape derivative based on the cost functional with pde constraints defined on the continuous level. The second shape derivative is similar but using a discretized cost functional that allows for the embedding of CutFEM formulations directly in the formulation. Different from the first two methods, the third shape derivative is based on a discrete formulation where perturbations of the domain are built into the variational formulation on the unperturbed domain. This is realized by using the so-called boundary value correction method that was originally introduced to allow for high order approximations to be realized using low order approximation of the domain. The theoretical discussion is illustrated with a series of numerical examples showing that all three approaches produce similar result on the proposed Bernoulli problem.

Hyperbolic Solutions to Bernoulli’s Free Boundary Problem

Bernoulli's free boundary problem is an overdetermined problem in which one seeks an annular domain such that the capacitary potential satisfies an extra boundary condition. There exist two different types of solutions called elliptic and hyperbolic solutions. Elliptic solutions are ``stable'' solutions and tractable by variational methods and maximum principles, while hyperbolic solutions are ``unstable'' solutions of which the qualitative behavior is less known. We introduce a new implicit function theorem based on the parabolic maximal regularity, which is applicable to problems with loss of derivatives. Clarifying the spectral structure of the corresponding linearized operator by harmonic analysis, we prove the existence of foliated hyperbolic solutions as well as elliptic solutions in the same regularity class.

A second-order shape optimization algorithm for solving the exterior Bernoulli free boundary problem using a new boundary cost functional

The exterior Bernoulli problem is rephrased into a shape optimization problem using a new type of objective function called the Dirichlet-data-gap cost function which measures the $$L^2$$ -distance between the Dirichlet data of two state functions. The first-order shape derivative of the cost function is explicitly determined via the chain rule approach. Using the same technique, the second-order shape derivative of the cost function at the solution of the free boundary problem is also computed. The gradient and Hessian informations are then used to formulate an efficient second-order gradient-based descent algorithm to numerically solve the minimization problem. The feasibility of the proposed method is illustrated through various numerical examples.

On the existence of non-flat profiles for a Bernoulli free boundary problem

Abstract In this paper, we consider a large class of Bernoulli-type free boundary problems with mixed periodic-Dirichlet boundary conditions. We show that solutions with non-flat profile can be found variationally as global minimizers of the classical Alt–Caffarelli energy functional.

Bernoulli Free Boundary Problem for the Infinity Laplacian

We study the interior Bernoulli free boundary problem for the infinity Laplacian. Our results cover the existence, uniqueness, and characterization of solutions (above a threshold representing the ...

On the behavior of the free boundary for a one-phase Bernoulli problem with mixed boundary conditions

This paper is concerned with the study of the behavior of the free boundary for a class of solutions to a two-dimensional one-phase Bernoulli free boundary problem with mixed periodic-Dirichlet boundary conditions. It is shown that if the free boundary of a symmetric local minimizer approaches the point where the two different conditions meet, then it must do so at an angle of \begin{document}$ \pi/2 $\end{document} .

Homogenization of a Singular Perturbation Problem

We discuss homogenization of the singular perturbation problem $$ {\Delta}_p{u}_{\delta}^{\varepsilon }={f}^{\varepsilon }{\beta}_{\delta}\left({u}_{\delta}^{\varepsilon}\right)\kern1em in\kern0.5em {\mathrm{\mathbb{R}}}^n\backslash \overline{B_1} $$ with a constant boundary value on the ball. Here, Δp is the usual p-Laplacian operator. It is generally understood that the two parameters δ and e are in competition and two different behaviors may be exhibited, depending on which parameter tends to zero faster. We consider one scenario where we assume that e, the homogenization parameter, tends to zero faster than δ, the singular perturbation parameter. We show that there is a universal speed for which the limit solves a standard Bernoulli free boundary problem.

On the supremal version of the Alt–Caffarelli minimization problem

Abstract This is a companion paper to our recent work [Bernoulli free boundary problem for the infinity Laplacian, preprint 2018, https://arxiv.org/abs/1804.08573]. Here we consider its variational side, which corresponds to the supremal version of the Alt–Caffarelli minimization problem.

A new energy-gap cost functional approach for the exterior Bernoulli free boundary problem

The solution to a free boundary problem of Bernoulli type, also known as Alt-Caffarelli problem, is studied via shape optimization techniques. In particular, a novel energy-gap cost functional approach with a state constraint consisting of a Robin condition is proposed as a shape optimization reformulation of the problem. Accordingly, the shape derivative of the cost is explicitly determined, and using the gradient information, a Lagrangian-like method is used to formulate an efficient boundary variation algorithm to numerically solve the minimization problem. The second order shape derivative of the cost is also computed, and through its characterization at the solution of the Bernoulli problem, the ill-posedness of the shape optimization formulation is proved. The analysis of the proposed formulation is completed by addressing the existence of optimal solution of the shape optimization problem and is accomplished by proving the continuity of the solution of the state problems with respect to the domain. The feasibility of the newly proposed method and its comparison with the classical energy-gap type cost functional approach is then presented through various numerical results. The numerical exploration issued in the study also includes results from a second-order optimization procedure based on a Newton-type method for resolving such minimization problem. This computational scheme put forward in the paper utilizes the Hessian information at the optimal solution and thus offers a state-of-the-art numerical approach for solving such free boundary problem via shape optimization setting.

Shape optimization approach for solving the Bernoulli problem by tracking the Neumann data: A Lagrangian formulation

The exterior Bernoulli free boundary problem is considered and reformulated into a shape optimization setting wherein the Neumann data is being tracked. The shape differentiability of the cost functional associated with the formulation is studied, and the expression for its shape derivative is established through a Lagrangian formulation coupled with the velocity method. Also, it is illustrated how the computed shape derivative can be combined with the modified $H^1$ gradient method to obtain an efficient algorithm for the numerical solution of the shape optimization problem.

A cut finite element method for the Bernoulli free boundary value problem

We develop a cut finite element method for the Bernoulli free boundary problem. The free boundary, represented by an approximate signed distance function on a fixed background mesh, is allowed to intersect elements in an arbitrary fashion. This leads to so called cut elements in the vicinity of the boundary. To obtain a stable method, stabilization terms are added in the vicinity of the cut elements penalizing the gradient jumps across element sides. The stabilization also ensures good conditioning of the resulting discrete system. We develop a method for shape optimization based on moving the distance function along a velocity field which is computed as the H1 Riesz representation of the shape derivative. We show that the velocity field is the solution to an interface problem and we prove an a priori error estimate of optimal order, given the limited regularity of the velocity field across the interface, for the velocity field in the H1 norm. Finally, we present illustrating numerical results.