Existence of variational solutions to doubly nonlinear systems in nondecreasing domains

Technical Summary: Existence of Variational Solutions to Doubly Nonlinear Systems in Nondecreasing Domains

Problem Statement
The paper addresses the Cauchy–Dirichlet problem for doubly nonlinear parabolic systems of the form:
$$\partial_t (|u|^{q-1}u) - \text{div}(D_\xi f(x, u, Du)) = -D_u f(x, u, Du)$$
defined in a bounded, noncylindrical domain $E \subset \mathbb{R}^{n+1}$ (where $n \ge 2$). The domain $E$ is assumed to be nondecreasing with respect to time, meaning the time slices $E_t$ satisfy $E_s \subset E_t$ for $0 \le s \le t < T$. The parameters satisfy $q \in (0, \infty)$ and $p \in (1, \infty)$.

The integrand $f(x, u, \xi)$ is assumed to be:

1. Integrable in $x$ for fixed $(u, \xi)$.
2. Convex in $(u, \xi)$.
3. Satisfying a $p$-coercivity condition: $\nu|\xi|^p \le f(x, u, \xi)$.
   Crucially, the authors do not impose a specific upper growth condition on $f$ for the primary existence result, allowing for exponential or non-standard growth.

Methodology
The proof of existence relies on a nonlinear version of the method of minimizing movements (also known as the implicit Euler scheme or De Giorgi's minimizing movements). The approach combines techniques from previous works on doubly nonlinear equations in cylindrical domains (Böglein et al. [17]) and parabolic systems in noncylindrical domains (Böglein et al. [19]).

The procedure involves:

1. Time Discretization: The time interval $(0, T)$ is divided into steps of size $h_\ell = T/\ell$.
2. Iterative Minimization: At each time step, a sequence of functions $u_{\ell, i}$ is constructed by minimizing an elliptic variational functional:
   $$F_\ell[w; u_{\ell, i-1}] = \int_\Omega f(x, w, Dw) \, dx + \frac{1}{h_\ell} \int_\Omega b[u_{\ell, i-1}, w] \, dx$$
   where $b[u, v]$ is a specific boundary term related to the convexity of the power function $|v|^{q+1}$.
3. Energy Estimates and Convergence: The authors derive uniform energy estimates for the piecewise constant interpolations $u_\ell$. Using compactness arguments (specifically Lemma 3.11 involving the convergence of $|u_\ell|^{(q+1)/2}$), they establish the existence of a limit map $u$.
4. Handling Noncylindrical Domains: A key technical aspect is managing the time-varying domain. The nondecreasing nature of $E$ allows the authors to extend comparison maps and utilize the assumption $L^{n+1}(\partial E) = 0$ to ensure the limit map satisfies the correct lateral boundary conditions.
5. Variational Inequality: The limit map is shown to satisfy a variational inequality, which serves as the definition of a variational solution in this context.

Key Contributions and Results

1. Existence in Nondecreasing Domains (Theorem 2.3):
   The paper proves the existence of a variational solution $u \in C^0([0, T]; L^{q+1}(\Omega, \mathbb{R}^N)) \cap V^p_q(E)$.

   • Generality: Unlike previous results restricted to cylindrical domains or non-negative solutions, this result applies to general noncylindrical domains that are nondecreasing in time and allows for signed or vector-valued solutions.
   • Weak Assumptions on $f$: The existence proof holds without an upper growth condition on $f$, relying only on convexity and coercivity.
   • Regularity: The solution possesses a weak time derivative for the term $|u|^{q-1}u$ in a specific dual space, and the solution is continuous in time with respect to the $L^{q+1}$-norm.

2. Existence of Time Derivative in Dual Space (Theorem 2.5):
   Under additional assumptions—specifically that the domain satisfies a measure density condition (related to the complement being uniformly $p$-fat) and that $f$ satisfies a standard $p$-growth condition—the authors prove that the term $|u|^{q-1}u$ admits a distributional time derivative in the dual space $(V^{p,0}(E))'$.

3. Continuity in Time:
   The authors establish that any variational solution is continuous in time with values in $L^{q+1}(\Omega, \mathbb{R}^N)$. This is achieved by testing the variational inequality with a time-mollification of the solution, a technique adapted to the noncylindrical setting.

Significance and Claims
The paper positions itself as a generalization of existing theories for doubly nonlinear equations.

• Extension of Scope: It extends the existence theory from cylindrical domains to noncylindrical (specifically nondecreasing) domains, addressing a gap in the literature where such geometric variations were not fully treated for doubly nonlinear systems.
• Relaxed Growth Conditions: By removing the upper growth condition on the integrand $f$ for the existence result, the authors allow for a broader class of nonlinearities, including those with exponential growth, which were previously handled only under stricter constraints or in specific contexts.
• Methodological Synthesis: The work successfully synthesizes the "minimizing movements" approach for doubly nonlinear equations with the geometric handling of time-dependent domains.
• Uniqueness: The authors explicitly note that uniqueness is not established in this work. They cite that uniqueness for such systems is a delicate issue, particularly in noncylindrical domains, and remains an open question for the general case considered here, though it is known for specific cases like the porous medium equation with non-negative solutions.

The paper concludes that the methods developed can potentially be adapted for time-dependent boundary values, though the current work focuses on time-independent lateral boundary values to establish the core existence theory.