Cauchy-Dirichlet Problem for Linear Parabolic Equations

• The Cauchy-Dirichlet problem is a key boundary-initial value problem modeling heat-type evolution under Dirichlet conditions.
• Advancements utilize Sobolev, Morrey, and weighted spaces alongside singular integral methods to achieve global regularity and uniqueness.
• Modern extensions address degenerate, variable-coefficient, and fractional derivative cases, ensuring robust solutions in irregular domains.

The Cauchy-Dirichlet problem for linear parabolic equations is a central boundary-initial value problem in PDE theory, governing the evolution of solutions to second-order parabolic PDEs under prescribed initial and Dirichlet boundary conditions. These problems encompass classical heat-type equations in both divergence and non-divergence form and extend to degenerate, variable-coefficient, and fractional time-derivative cases, regularly appearing in analysis, mathematical physics, and probability. Modern advancements address global regularity, existence, uniqueness, and fine boundary behavior under minimal geometric and coefficient regularity assumptions, utilizing function spaces such as Sobolev, Morrey, and weighted Lebesgue spaces, parabolic measure theory, and singular integral methods.

1. Formulation and Function-Space Framework

Consider a bounded domain $S\subset\mathbb{R}^n$ of class $C^{1,1}$ and $Q = S \times (0, T)$, with $T>0$. The Cauchy-Dirichlet problem for a linear parabolic equation in non-divergence form is

$$\begin{cases} u_t(x, t) - a^{ij}(x, t) D_{ij} u(x, t) = f(x, t), & \text{a.e. } (x, t) \in Q, \ u(x, 0) = 0, & x \in S, \ u(x, t) = 0, & (x, t) \in \partial S \times (0, T), \end{cases}$$

where $D_{ij}u = \partial^2 u / \partial x_i \partial x_j$ and the solution is sought in the Sobolev–Morrey space $W^{2,1}_p(Q)$ or its generalizations. The parabolic boundary $\partial_p Q$ consists of $S \times \{0\}$ (the initial face) and $\partial S \times [0, T]$ (the lateral boundary). Coefficient regularity assumptions typically impose $a^{ij}(\cdot) \in L^\infty(Q)$, uniform ellipticity, and, in advanced theory, vanishing mean oscillation ($VMO(Q)$), thus ensuring the capacity to treat discontinuous coefficients (Guliyev et al., 2012).

Generalizations address divergence-form equations, time- and space-varying domains (admissible for the parabolic geometry), and boundary data in various function spaces such as generalized Morrey spaces $M^{p, \phi}(Q)$, for which specific growth and integrability conditions on $\phi$ are imposed. The relevant solution classes, non-tangential maximal function frameworks, and weighted spaces enable treatment of solutions with singular data and domains exhibiting minimal smoothness (Dindoš et al., 2014, Hara, 6 Jan 2026).

2. Regularity: Morrey, Weighted, and Sobolev Estimates

Sharp regularity results depend on both spatial and data regularity. A key advancement is the derivation of global a priori estimates in generalized parabolic Morrey spaces (Guliyev et al., 2012): $$\|u\|_{W^{2,1}_{p, \phi}(Q)} := \sum_{2\ell + |\alpha| \leq 2} \|D_t^\ell D_x^\alpha u\|_{p, \phi; Q} \leq C \|f\|_{p, \phi; Q},$$ where $C$ depends on domain and coefficient bounds and the $VMO$-modulus. Analogous weighted norm estimates in anisotropic $W^{2,1}_{p, q; \alpha}(\Omega \times (0, T))$ are established for wedges and domains with edges/corners; these exploit weighted Sobolev spaces alongside critical exponent techniques (Kozlov et al., 2011). The global Hölder solvability extends to extremely rough domains satisfying the capacity density condition (CDC), utilizing barrier arguments to provide quantitative boundary regularity even for right-hand sides $f$ exhibiting near-critical singularities (Hara, 6 Jan 2026).

For nontrivial Dirichlet data or more singular scenarios (e.g., white noise boundary conditions), solutions are constructed in weighted $L^p$-spaces with the distance-to-boundary as weight (Goldys et al., 2021). Solutions remain well-posed and admit fine regularity up to stochastic Markovian properties.

3. Singular Integral and Operator-Theoretic Methods

Continuity and boundedness of parabolic Calderón–Zygmund operators and their commutators are instrumental in establishing regularity in generalized Morrey spaces. If $T$ is a Calderón–Zygmund operator with kernel satisfying appropriate homogeneity and cancellation, and $\phi$ satisfies

$$\int_r^\infty \phi(x, s) s^{-(n+2)/p - 1} ds \leq C \phi(x, r),$$

then $T$ is bounded on $M^{p, \phi}$, and analogous statements hold for commutators $[a, T]$, with $a \in BMO$ (Guliyev et al., 2012). The proof techniques exploit decomposition lemmas (splitting $f$ into local and nonlocal parts), Hardy-operator inequalities critical to the function-space setup, and John–Nirenberg inequalities for $BMO/VMO$ control. Representation formulas for derivatives of the solution involve singular integrals acting on the data and coefficients, critical to transferring known $L^p$-regularity to weighted and Morrey frameworks.

Boundary representation necessitates “reflection techniques,” enabling the passage from singular to nonsingular integral representations, particularly in half-space or near-boundary analysis. This analytic machinery underpins the regularity theory both locally and globally (Guliyev et al., 2012).

4. Weak Solution, Measure, and Representation Theory

Weak solutions are defined variationally. Given $A(x, t)$ uniformly elliptic and $f \in L^2$, the solution minimizes an associated convex functional subject to the PDE constraint, as in the Brezis–Ekeland–Armstrong–Mourrat variational approach to (possibly degenerate) kinetic Fokker–Planck operators (Litsgård et al., 2020). The boundary value problem is formulated with respect to adapted function spaces reflecting the underlying anisotropies.

Solutions admit representation via parabolic (caloric) measure: for fixed $(x, t)$, if $g$ is the continuous boundary data, the solution takes the form

$$u(x, t)=\int_{\partial Q} g(y, s) d\omega^{x, t}(y, s),$$

where $\omega^{x, t}$ is the parabolic measure associated to the operator. Under bounded, measurable, and elliptic coefficients (not necessarily symmetric), the measure is $A_\infty$-absolutely continuous with respect to Lebesgue surface measure, ensuring reverse Hölder properties of the Poisson kernel and broad solvability in $L^p$-spaces (Auscher et al., 2016, Dindoš et al., 2014).

For divergence-form parabolic equations, existence, uniqueness, and non-tangential maximal function solvability in $L^p$ follow from the absolute continuity and doubling of parabolic measure under small Carleson-norm hypotheses on $A$ (Dindoš et al., 2014). For time- or space-discontinuous or degenerate coefficients, the measure theory has been extended to allow even rougher settings (Litsgård et al., 2020).

5. Minimal Regularity, Duality, and Carleson/VMO Conditions

The modern theory achieves solvability and regularity of the Cauchy–Dirichlet problem under nearly optimal minimal smoothness on both the coefficients and the boundary:

• VMO coefficients: Global $W^{2,1}_p$ and generalized Morrey estimates hold for non-divergence form equations as soon as coefficients are $VMO$ in $Q$ (Guliyev et al., 2012).
• Carleson measure conditions: For divergence form, the small Carleson-norm hypothesis on $\nabla_x A$, $\partial_t A$ allows $L^p$ boundary data and ensures $A_\infty$ properties for the parabolic measure, even with non-symmetric coefficients (Dindoš et al., 2014, Auscher et al., 2016).
• Fractional time derivatives: Regularity theory extends to Caputo-type time-fractional derivatives $\mathbb{D}_t^\alpha$ with $0 < \alpha < 2$, with maximal continuity/Hölder regularity in time and space and compatibility conditions at $t=0$ (Guidetti, 2018).

Duality theorems assert that solvability of the Regularity problem $(R)_p$ for the forward equation implies solvability of the adjoint Dirichlet problem $(D^*)_{p'}$ for the backward (adjoint) operator. These results extend the Kenig–Pipher elliptic duality into the parabolic regime and provide a precise functional-analytic correspondence between primal and adjoint boundary behaviors (Dindoš et al., 2017).

6. Domains with Singularities and Geometric Flexibility

Global existence and regularity have been established on domains much rougher than classical smooth settings:

• Capacity Density Condition (CDC): Global Hölder continuous solutions exist for divergence-form equations under CDC on the boundary, accommodating nearly critical right-hand side singularities, extending results beyond Lipschitz and NTA domains (Hara, 6 Jan 2026).
• Wedges and edges: Weighted coercive estimates in $L_{p, q}^\alpha$ and anisotropic Sobolev spaces provide existence and regularity up to corners, conical points, and edges. The concept of a “critical exponent” $\lambda_c$ dictates the appropriate interval of weight exponents for unique solvability (Kozlov et al., 2011).

A global barrier function of Ancona-type, decaying like a boundary distance to some power, is pivotal in these constructions. Such barriers allow comparison principles and boundary modulus of continuity even when solutions or data exhibit singular near-boundary behavior.

7. Extensions and Further Applications

The aggregated theory encompasses:

• Stochastic time/space Dirichlet data (e.g., white noise), with solution frameworks in weighted Sobolev spaces and interpretations as infinite-dimensional Ornstein–Uhlenbeck processes (Goldys et al., 2021).
• Degenerate and Kolmogorov-Fokker-Planck type operators with structurally adapted variational spaces and non-Euclidean Sobolev frameworks (Litsgård et al., 2020).
• Fractional-in-time evolution, via Caputo derivatives, with a comprehensive maximal regularity theory in both classical and mixed Hölder spaces, essential for quasilinear and semilinear parabolic PDE approaches (Guidetti, 2018).

These advances unify and extend classical $L^p$- and Morrey-space theory, allow for modeling of non-smooth media (discontinuous/rough coefficients), and establish robust a priori and boundary estimates using singular integral, measure-theoretic, and variational methods (Guliyev et al., 2012, Dindoš et al., 2014, Hara, 6 Jan 2026).

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References:

• "Generalized Morrey regularity for parabolic equations with discontinuity data" (Guliyev et al., 2012)
• "The Dirichlet boundary problem for second order parabolic operators satisfying Carleson condition" (Dindoš et al., 2014)
• "Global Hölder Solvability of parabolic equations on domains with capacity density conditions" (Hara, 6 Jan 2026)
• "The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients in a wedge" (Kozlov et al., 2011)
• "On maximal regularity for the Cauchy-Dirichlet mixed parabolic problem with fractional time derivative" (Guidetti, 2018)
• "Linear parabolic equation with Dirichlet white noise boundary conditions" (Goldys et al., 2021)
• "The Dirichlet problem for Kolmogorov-Fokker-Planck type equations with rough coefficients" (Litsgård et al., 2020)
• "The Dirichlet problem for second order parabolic operators in divergence form" (Auscher et al., 2016)
• "Parabolic Regularity and Dirichlet boundary value problems" (Dindoš et al., 2017)