On the Viscosity Solutions of Parabolic p-Laplacian Equations with Capillary-Type Boundary Conditions

Abstract: In this paper, we establish the well-posedness and large-time asymptotic behavior of viscosity solutions to singular/degenerate parabolic $p$-Laplacian equations with general capillary-type boundary conditions, including Neumann and prescribed contact angle cases, on strictly convex domains. By establishing a gradient estimate independent of the $C<sup>0$ norm of the solution via the maximum principle, and by analyzing the problem through an approximation procedure together with associated elliptic eigenvalue problems, we prove the existence, uniqueness, and asymptotic behavior of solutions. For the elliptic problem with Neumann boundary conditions, we first focus on flat domains with the zero Neumann condition. By reflecting $u$ across the flat boundary $T_1$ and then using inf- and sup-convolution arguments in the reflected domain, we obtain the $C<sup>{1,α}$ result. For the general elliptic case, we obtain sharp global $C<sup>{1,α}$ regularity by flattening the boundary and employing compactness arguments together with an ``improvement of flatness'' iteration. With an extra condition in the iteration, we can also deal with the singular case $1<p<2$. In the parabolic setting, the spatial Hölder regularity of $Du$ follows from elliptic estimates combined with the Lipschitz continuity of $u$ in time, which in turn yields joint Hölder continuity in $(x,t)$. Extensions to non-convex domains are also discussed by incorporating a suitable forcing term.

• The paper introduces an a priori gradient estimate for viscosity solutions of p-Laplacian equations under capillary-type boundary conditions.
• It develops a robust approximation method ensuring existence, uniqueness, and optimal regularity in both singular and degenerate regimes.
• The study characterizes large-time asymptotics via a nonlinear eigenvalue, extending results to non-convex domains with additional forcing.

Viscosity Solutions for Parabolic $p$-Laplacian Equations with Capillary-Type Boundary Conditions

Introduction and Problem Setting

This paper addresses the well-posedness, regularity properties, and large-time asymptotics for viscosity solutions of parabolic $p$-Laplacian equations with general capillary-type boundary conditions on strictly convex (and, with modifications, non-convex) domains. The model problem is

$$\begin{cases} u_t = \operatorname{div}(|Du|^{p-2}Du) + f(x, u) & \text{in } \Omega \times [0, \infty), \ |Du|^{q-1} u_\nu = -\phi(x) & \text{on } \partial\Omega \times [0, \infty), \ u(x,0) = u_0(x) & \text{in } \Omega, \end{cases}$$

with $p > 1$, $q \geq 0$, and with boundary conditions covering Neumann ($q=1$), conormal ($q=p-1$), and prescribed contact angle ($q=0$) situations. The analysis includes both singular ($1 < p < 2$) and degenerate ($p>2$) regimes, as well as essential conditions on the data and the geometry of the domain. For strictly convex domains the authors establish existence, uniqueness, optimal regularity, and long-time behavior, while treatment for non-convex domains is enabled through an additional forcing term.

Methodological Innovations

A key contribution is an a priori gradient estimate, independent of the $p$0 norm, obtained via a refined maximum principle argument. This estimate is robust under singular and degenerate settings and is valid for general capillary-type boundary conditions, including the challenging non-linear case $p$1. The estimate is crucial for compactness arguments and for establishing uniform parabolicity in approximation procedures.

The authors employ an approximation through regularized problems with parameter $p$2, passing to the limit to recover results for the original singular/degenerate problems. They systematically construct strong barriers for both interior and boundary estimates that enable the control of both classical and viscosity solutions, leveraging strictly convex geometry.

On flat and $p$3 domains, sharp $p$4 regularity up to the boundary, independently of $p$5, is achieved. The arguments for the Neumann case do not directly follow from classical viscosity or weak theory due to the $p$6 dependence; instead, the authors employ reflection techniques and inf/sup-convolutions in the reflected domain, supported by recent advances in the equivalence between viscosity and weak solutions.

For non-convex domains, following and extending ideas from the mean curvature flow literature, they add a high-order forcing term, ensuring that critical gradient estimates can still be obtained. The non-convex analysis thus mimics the structure of the convex case in a generalized context.

Main Results

Existence and Regularity

• For strictly convex domains and $p$7, there exists a unique viscosity solution which is globally Lipschitz in $p$8, and $p$9 in $$\begin{cases} u_t = \operatorname{div}(|Du|^{p-2}Du) + f(x, u) & \text{in } \Omega \times [0, \infty), \ |Du|^{q-1} u_\nu = -\phi(x) & \text{on } \partial\Omega \times [0, \infty), \ u(x,0) = u_0(x) & \text{in } \Omega, \end{cases}$$0, with optimal exponent $$\begin{cases} u_t = \operatorname{div}(|Du|^{p-2}Du) + f(x, u) & \text{in } \Omega \times [0, \infty), \ |Du|^{q-1} u_\nu = -\phi(x) & \text{on } \partial\Omega \times [0, \infty), \ u(x,0) = u_0(x) & \text{in } \Omega, \end{cases}$$1 dictated by the best-known interior and boundary regularity for the $$\begin{cases} u_t = \operatorname{div}(|Du|^{p-2}Du) + f(x, u) & \text{in } \Omega \times [0, \infty), \ |Du|^{q-1} u_\nu = -\phi(x) & \text{on } \partial\Omega \times [0, \infty), \ u(x,0) = u_0(x) & \text{in } \Omega, \end{cases}$$2-Laplacian. The results are explicit for Neumann and conormal conditions, matching or generalizing existing literature.
• The paper delivers existence and uniqueness for solutions with prescribed contact angle ($$\begin{cases} u_t = \operatorname{div}(|Du|^{p-2}Du) + f(x, u) & \text{in } \Omega \times [0, \infty), \ |Du|^{q-1} u_\nu = -\phi(x) & \text{on } \partial\Omega \times [0, \infty), \ u(x,0) = u_0(x) & \text{in } \Omega, \end{cases}$$3) under a near-verticality condition for the angle data $$\begin{cases} u_t = \operatorname{div}(|Du|^{p-2}Du) + f(x, u) & \text{in } \Omega \times [0, \infty), \ |Du|^{q-1} u_\nu = -\phi(x) & \text{on } \partial\Omega \times [0, \infty), \ u(x,0) = u_0(x) & \text{in } \Omega, \end{cases}$$4. Here all estimates and methods extend except in the absence of strong obliqueness, which presents intrinsic complications.
• For non-convex domains, with an appropriate choice of the forcing term, the full range of existence and regularity results is recovered without the strict convexity hypothesis.

Large-Time Asymptotics

• Uniform convergence to translating profiles: In the uniformly parabolic regularized system, the solution $$\begin{cases} u_t = \operatorname{div}(|Du|^{p-2}Du) + f(x, u) & \text{in } \Omega \times [0, \infty), \ |Du|^{q-1} u_\nu = -\phi(x) & \text{on } \partial\Omega \times [0, \infty), \ u(x,0) = u_0(x) & \text{in } \Omega, \end{cases}$$5 admits a profile for large time; the difference between any two solutions with smooth initial data vanishes in $$\begin{cases} u_t = \operatorname{div}(|Du|^{p-2}Du) + f(x, u) & \text{in } \Omega \times [0, \infty), \ |Du|^{q-1} u_\nu = -\phi(x) & \text{on } \partial\Omega \times [0, \infty), \ u(x,0) = u_0(x) & \text{in } \Omega, \end{cases}$$6 as $$\begin{cases} u_t = \operatorname{div}(|Du|^{p-2}Du) + f(x, u) & \text{in } \Omega \times [0, \infty), \ |Du|^{q-1} u_\nu = -\phi(x) & \text{on } \partial\Omega \times [0, \infty), \ u(x,0) = u_0(x) & \text{in } \Omega, \end{cases}$$7.
• Asymptotic speed selection via eigenproblems: The large-time behavior is governed by a unique eigenvalue $$\begin{cases} u_t = \operatorname{div}(|Du|^{p-2}Du) + f(x, u) & \text{in } \Omega \times [0, \infty), \ |Du|^{q-1} u_\nu = -\phi(x) & \text{on } \partial\Omega \times [0, \infty), \ u(x,0) = u_0(x) & \text{in } \Omega, \end{cases}$$8 of an associated capillary-type elliptic eigenproblem. For the degenerate/singular limit, the viscosity solutions satisfy

$$\begin{cases} u_t = \operatorname{div}(|Du|^{p-2}Du) + f(x, u) & \text{in } \Omega \times [0, \infty), \ |Du|^{q-1} u_\nu = -\phi(x) & \text{on } \partial\Omega \times [0, \infty), \ u(x,0) = u_0(x) & \text{in } \Omega, \end{cases}$$9

with an explicit, optimal rate.

• Convergence to constants in the zero data case: With zero boundary and reaction data, solutions converge to a spatial constant, linked to the null eigenfunction of the limiting elliptic problem.

Boundary Regularity

For the Neumann and prescribed contact angle problems, global Hölder continuity of both the solution and its gradient to the boundary is established, via improvement of flatness iterations integrally combined with sup/inf-convolution techniques and the equivalence of viscosity and weak solution concepts.

Implications

Analytical PDE and Viscosity Solution Theory

The work pushes forward the theory of singular/degenerate fully nonlinear parabolic equations, in particular with boundary conditions that go beyond the conormal or homogeneous Neumann case. The technical developments, including a gradient bound independent of the $p > 1$0 norm and the translation of boundary regularity results from the weak to the viscosity setting, fill gaps in the literature---notably covering cases previously inaccessible via standard oblique boundary theory.

Geometric Flows and Capillarity

The analytical treatment applies to non-parametric geometric flows with capillarity, including mean curvature and level-set flows subject to complex boundary conditions. The gradient bounds and barrier constructions rely crucially on domain geometry; their adaptation to the convex/non-convex dichotomy will be useful in future geometric PDE developments. The capillary-type boundary conditions model physical phenomena with contact lines and interfaces, relevant for mathematical physics and materials science.

Future Directions

• Extension to less regular domains: While the strictly convex case is thoroughly addressed, the methods invite further exploration for general Lipschitz domains, possibly with weaker regularity, or with minimal geometric constraints.
• Improvement of large-time profile characterization: The asymptotic behavior, linked to a nonlinear eigenvalue, hints at further spectral and variational approaches, particularly for explicit determination of rates and profiles in time-dependent capillarity problems.
• Stochastic PDEs and nonlocal/nonlinear fluxes: The paradigm of barrier/multiplier construction and viscosity/weak solution equivalence is translatable to other classes of nonlinear PDEs, e.g., nonlocal or stochastic PDEs with capillary-type effects.

Conclusion

The paper provides a comprehensive analysis for viscosity solutions of parabolic $p > 1$1-Laplacian equations with capillary-type boundary conditions, unifying singular, degenerate, and general oblique geometries. The main technical advances are in gradient estimates, regularity up to the boundary, and the characterization of large-time behavior via nonlinear eigenvalue problems. The results resolve well-posedness issues for a rich class of capillarity-driven evolutions, setting a foundation for further theoretical and applied developments in nonlinear PDEs with complex boundary interactions.