Area minimizing hypersurfaces modulo p: a geometric free-boundary problem

We study the one-phase Alt-Phillips free boundary problem, focusing on the case of negative exponents . The goal of this paper is twofold. On the one hand, we prove smoothness of -regular free boundaries by reducing the problem to a class of degenerate quasilinear PDEs, for which we establish Schauder estimates. Such a method provides a unified proof of the smoothness for general exponents. On the other hand, by exploiting the higher regularity of solutions, we derive a new stability condition for the Alt-Phillips problem in the negative exponent regime, ruling out the existence of nontrivial axially symmetric stable cones in low dimensions. Finally, we provide a variational criterion for the stability of cones in the Alt-Phillips problem, which recovers the one for minimal surfaces in the singular limit as .

Abstract We study fully nonlinear uniformly elliptic equations with logarithmic singular absorption terms. In the absence of scaling properties, we establish the existence of solutions and characterize their optimal growth along the free boundary and sharp local regularity. Additionally, we derive non-degeneracy results and provide finer geometric estimates for the free boundary.

A game approach to free boundary problems of anisotropic forced mean curvature flow equations

Stable cones in the Alt-Phillips free boundary problem

A free boundary problem with delay-induced nonlocal impulsive reaction

Analysis of a free boundary problem modeling immune response to fungal infection in multi-space dimensional case

Recent work (Raufaste et al. 2022 Soft Matter , vol. 18, p. 4944) studied the dynamics of a soap film in the shape of an unstable minimal surface whose evolution is governed in part by the frictional forces associated with surface Plateau border (SPB) motion. In this note, we study a variant of this problem in which a half-catenoid bounded by a wire loop and a fluid bath axisymmetrically surrounds a cylindrical rod with a radius equal to the neck of the critical catenoid given by the wire loop. When the half-catenoid is brought just beyond the point of instability, the film touches the cylinder and separates from the bath, creating an SPB that is dragged upwards along the rod by the now unstable soap film, and asymptotically relaxes to a new stable annular minimal surface. For this free-boundary problem involving an unstable initial condition, we find the dynamics by balancing the capillary force of successive unstable minimal surfaces spanning the SPB and the wire loop with the frictional force associated with the moving SPB. We find good agreement between theory and experiment using the frictional force $f\sim \textit{Ca}^{2/3}$ given by Bretherton’s law, where $ \textit{Ca} $ is the capillary number.

We consider an initial–boundary value problem modeling in situ leaching of rare earths with a special periodic structure by an acid solution, improving some previous studies by the second author and collaborators. At the microscopic scale, fluid motion in the pore space is described by the Stokes equations for a slightly compressible fluid coupled with the deformation of the elastic skeleton, governed by the Lamé system, and the diffusion equation for the acid solution. Due to rock dissolution, the interface between liquid and solid phases is unknown (it is a free boundary) and must be determined as part of the solution. To overcome this difficulty, we introduce a family of approximate microscopic models with prescribed pore geometry and establish their well-posedness in a weak formulation. Using a priori estimates and Galerkin’s method, we obtain existence results and apply the method of two-scale convergence for periodic structures to derive the corresponding homogenized macroscopic model. Finally, a fixed-point argument yields existence and uniqueness for the resulting macroscopic system.

ABSTRACT In this paper, we investigate a nonlocal diffusion SIR epidemic model with double free boundaries, where the infection rate and the recovery rate have been affected by media coverage and the number of hospital beds, respectively. We first give the existence and uniqueness of a global solutions, and then study the long‐time behaviors and sufficient conditions for the disease spreading and vanishing.

We study the one-dimensional capillary-driven retraction of a finite planar liquid sheet in the asymptotic regime where both the initial length-to-thickness ratio l 0 / h 0 and a characteristic Ohnesorge number, Oh, which is inversely related to the Reynolds number of the flow, are large. In this regime, the fluid domain decomposes into two regions: a thin-film region governed by one-dimensional mass and momentum equations and a small tip region near the free edge described by a self-similar Stokes flow. Asymptotic matching between these regions yields an effective boundary condition for the thin-film region, representing a balance between viscous and capillary forces at the free edge. We find that surface tension drives the thin-film flow only through this boundary condition, while the local momentum balance is dominated by viscous and inertial stresses. We show that the thin-film flow possesses a conserved quantity, reducing the governing equations to a one-dimensional free-boundary problem in which the sheet thickness satisfies the heat equation with time-dependent boundary conditions. The reduced problem depends on a single dimensionless parameter L = l 0 / ( 4 h 0 Oh ) . Numerical solutions of the reduced model agree well with previous studies based on different formulations, including full Navier-Stokes simulations, and reveal that the sheet undergoes distinct retraction regimes depending on L and a dimensionless time after rupture T . We derive asymptotic approximations for the thickness profile, velocity profile, and retraction speed during the early and late stages of retraction. At early times, the retraction speed grows as T 1 / 2 , characteristic of diffusive processes, while at late times it decays as 1 / T 2 due to the finiteness of the sheet. Both results are consistent with previous findings. We also analyze an intermediate regime, not previously studied in detail, which arises for very long sheets ( L ≫ 1 ). During this phase, the retraction speed approaches the classical Taylor-Culick value, despite the sheet not exhibiting a large circular rim as in classical studies of film rupture. When T ≈ L , the speed undergoes a sudden deceleration, which we characterize numerically, going from the Taylor-Culick speed to that predicted by the late-time asymptotics.

We derive a convex relaxation principle for a large class of non convex variational problems where the functional to be minimized involves a one homogeneous gradient energy. This applies directly to free boundary or multiphase problems in the case of the classical total variation or of some anisotropic variants. The underlying argument is an exclusion principle which states that any global minimizer avoids taking values in the intervals where the lower order potential is nonconvex. This allows using duality methods and deriving a saddle point characterization of the global minimizers. A numerical validation of our principle is presented in the case of several free boundary and multiphase problems that we treat through a primal-dual algorithm. The accuracy of the interfaces and the convergence of the algoritm benefit in a large way of a new epigraphical projection method that we introduced to tackle the non differentiability of the convexified Lagrangian.

Abstract We consider a toy model of rate-independent droplet motion on a surface with contact angle hysteresis based on the one-phase Bernoulli free boundary problem. We introduce a notion of solutions based on an obstacle problem. These solutions jump “as late and as little as possible”, a physically natural property that energy solutions do not satisfy. When the initial data is star-shaped, we show that obstacle solutions are uniquely characterized by satisfying the local stability and dynamic slope conditions. This is proved via a novel comparison principle, which is one of the main new technical results of the paper. In this setting we can also show the (almost) optimal $C^{1,1/2-}$ -spatial regularity of the contact line. This regularity result explains the asymptotic profile of the contact line as it de-pins via tangential motion similar to de-lamination. Finally we apply our comparison principle to show the convergence of minimizing movements schemes to the same obstacle solution, again in the star-shaped setting.

Two-phase free boundary problems for a class of fully nonlinear double-divergence systems

The Patlak–Keller–Segel system of equations (PKS) is a classical example of an aggregation-diffusion equation. It describes the aggregation of some organisms via chemotaxis, limited by some nonlinear diffusion. It is known that for some choice of this nonlinear diffusion, the PKS model asymptotically leads to phase separation and mean-curvature-driven free boundary problems. In this paper, we focus on the elliptic–parabolic PKS model and we obtain the first unconditional convergence result in dimensions 2 and 3 toward the volume-preserving mean curvature flow. This work builds up on previous results that were obtained under the assumption that phase separation does not cause energy loss in the limit. In order to avoid this assumption, we rely on the Brakke-type formulation of the mean curvature flow and a reinterpretation of the problem as an Allen–Cahn equation with a nonlocal forcing term.

We study a class of free-boundary problems for degenerate one-phase Alt–Caffarelli functionals J_{Q}(v, \Omega):= \int_{\Omega}|\nabla v|^{2} + Q^{2}(x)\chi_{\{v>0\}}dx . More specifically, we consider Q(x)= \mathop\mathrm{dist}\nolimits(x, \Gamma)^{\gamma} for an affine k -plane \Gamma and \gamma>0 . Because Q vanishes on \Gamma , the techniques of Alt and Caffarelli (1981) for proving non-degeneracy of local minimizers u and weak geometric regularity of their positivity sets \{u>0\} fail near \Gamma . In this note we prove that despite the degeneration of Q local minimizers, u and \{u>0\} still satisfy an analogous non-degeneracy condition near \Gamma . This non-degeneracy is sufficient to eliminate the existence of cusps in a wider class of cases than previously known.

Abstract We present a new three-parameter family of self-consistent equilibrium models for quasi-relaxed stellar systems that are subject to the combined action of external tides and rigid internal rotation. These models provide an idealised description of globular clusters that rotate asynchronously with respect to their orbital motion around a host galaxy. Model construction proceeds by extension of the truncated King models, using a newly defined asynchronicity parameter to couple the tidal and rotational perturbations. The method of matched asymptotic expansion is used to derive a global solution to the free boundary problem posed by the corresponding set of Poisson-Laplace equations. We explore the relevant parameter space and outline the intrinsic properties of the resulting models, both structural and kinematic. Their triaxial configuration, characterised by extension in the direction of the galactic centre and flattening toward the orbital plane, is found to depart further from spherical symmetry for larger values of the asynchronicity parameter. We hope that these simplified analytical models serve as useful tools for investigating the interplay of tidal and rotational effects, providing an equilibrium description that complements, and may serve as a basis for, more realistic numerical simulations.

Abstract We consider the free boundary problem for a 3-dimensional, incompressible, irrotational liquid drop of nearly spherical shape with capillarity. We study the problem from the beginning, extending some classical results from the flat case (capillary water waves) to the spherical geometry: the reduction to a problem on the boundary, its Hamiltonian structure, the analyticity and tame estimates for the Dirichlet-Neumann operator in Sobolev class, and a linearization formula for it, both with the method of the good unknown of Alinhac and by a geometric approach. Then, also thanks to the analyticity of the operators involved, we prove the bifurcation of traveling waves, which are nontrivial (i.e., nonspherical) fixed profiles rotating with constant angular velocity. To the best of our knowledge, this is the first example of global-in-time nontrivial solutions of the free boundary problem for the capillary liquid drop.

In this paper we are interested in the study of a two-phase problem equipped with the $\Phi$-Laplacian operator $$ \Delta_\Phi u \coloneqq \mbox{div} \bigg(\varphi(|\nabla u|) \dfrac{\nabla u}{|\nabla u|}\bigg), $$% where $\Phi(s)=e^{s^2}-1$ and $\varphi=\Phi'$. We obtain the existence, boundedness, and Log-Lipschitz regularity of the minimizers of the energy functional associated to the two-phase problem. Furthermore, we also prove that the free boundaries of these minimizers have locally finite perimeter and Hausdorff dimension at most $(N-1)$.

Abstract The vacuum free boundary problems of compressible flows are of great importance in the mathematical theory of fluid mechanics. In this paper, we continue the analysis of the compressible Navier–Stokes–Poisson equations with vacuum free boundary. We first show the global existence of axisymmetric strong solutions which extends a previous contribution of the authors (Li and Guo 2023 Calc. Var. PDE 62 109) by the method of perturbation analysis. We then show the nonlinear asymptotic stability with very detailed convergence rates, where some new energy functionals and truncation method are introduced. Particularly, it is shown that the derivatives of velocity components u φ and u z possess faster time decay rates than the one of u r . These results can apply to not only the white dwarfs but also the polytropic gases with adiabatic exponent γ ∈ ( 5 / 4 , ∞ ) , since a more general classes of pressure functions are considered (see (1.10) and (1.11)).