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This paper investigates the precise dependence of the Hölder exponent in partial C1,α regularity results for minimizers of variational integrals on the properties of zero-order terms, particularly focusing on Morrey-Hölder conditions, and applies these findings to sharpen Massari's regularity theorem for sets with variational mean curvature in Lp.

Abstract

Schmidt, T., & Schütt, J. H. (2024). Partial regularity for variational integrals with Morrey-H"older zero-order terms, and the limit exponent in Massari’s regularity theorem. *arXiv preprint arXiv:2410.03338v1*.

This paper aims to determine the sharp dependence of the Hölder exponent α in the partial C1,α regularity of minimizers for variational integrals on the structural assumptions imposed on the zero-order terms, specifically focusing on Morrey-Hölder conditions. Additionally, the research seeks to apply these findings to refine the regularity conclusion in Massari's regularity theorem for sets with variational mean curvature in Lp.

The authors employ the A-harmonic approximation method to establish partial regularity results. This involves deriving a Caccioppoli inequality, proving approximate A-harmonicity, and establishing excess decay estimates for local minimizers. The sharp dependence of the Hölder exponent on the Morrey-Hölder conditions is meticulously tracked throughout these steps. The application to Massari's theorem leverages the established results in the non-parametric setting and utilizes the connection between minimizers of specific variational integrals and sets of variational mean curvature.

- The paper provides a refined partial regularity theorem (Theorem 1.2) for variational integrals with Morrey-Hölder zero-order terms, explicitly demonstrating the sharp dependence of the Hölder exponent α on the parameters of the Morrey-Hölder condition.
- It establishes variants of the main theorem for a priori locally bounded minimizers (Theorem 1.3) and for minimizers with a priori gradient bounds in non-uniformly elliptic cases (Theorem 1.4).
- The research culminates in proving the optimal Hölder exponent in Massari's regularity theorem (Theorem 1.5), confirming the conjecture of [44, Remark 3.4] regarding the limit behavior of the exponent.

The study demonstrates the crucial role of Morrey-Hölder conditions in determining the regularity of minimizers for variational integrals. The precise dependence of the Hölder exponent on these conditions, as established in this work, provides a deeper understanding of the interplay between the structure of the integrand and the regularity of minimizers. The application to Massari's theorem showcases the strength of the obtained results and their potential to sharpen existing regularity theory in geometric analysis.

This research significantly contributes to the field of regularity theory for variational problems by providing a refined understanding of the role of zero-order terms and their impact on the regularity of minimizers. The explicit determination of the Hölder exponent's dependence on Morrey-Hölder conditions offers valuable insights for analyzing a wide range of variational models. Moreover, the sharpening of Massari's theorem has important implications for the study of minimal surfaces and related geometric problems.

The paper primarily focuses on the case of scalar-valued variational integrals. Extending the results to the fully vectorial setting with rank-one convexity assumptions on the integrand would be an interesting avenue for future research. Additionally, exploring the implications of the refined regularity results for specific applications in areas such as material science, image processing, and optimal control could lead to further advancements in these fields.

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Extending the results on the sharp dependence of the Hölder exponent to more general growth conditions on the integrand, going beyond the quadratic growth considered in the paper, presents a significant challenge and constitutes an active area of research in regularity theory. Here's a breakdown of potential approaches and challenges:
1. Growth Conditions and Function Spaces:
Superquadratic Growth (p-growth): For integrands with growth like $|z|^p$ for $p>2$, one needs to work with Sobolev spaces $W^{1,p}$ instead of $W^{1,2}$. The techniques often involve embedding theorems for these spaces and may require different approaches to Caccioppoli inequalities and the A-harmonic approximation.
Non-standard Growth: Even more challenging are cases where the integrand doesn't have a uniform polynomial growth rate, for example, integrands with variable exponents like $|z|^{p(x)}$. These require tools from the theory of variable exponent Lebesgue and Sobolev spaces, and the regularity theory is less developed.
2. Modified Techniques:
Caccioppoli Inequalities: Deriving suitable Caccioppoli inequalities, which are crucial for controlling the energy of the minimizer, becomes more involved. The test functions and manipulations used in the quadratic case might need to be adapted to the specific growth condition.
A-harmonic Approximation: The A-harmonic approximation lemma, a cornerstone of the method, relies on the properties of solutions to linear elliptic systems. For non-quadratic growth, one might need to consider generalizations of A-harmonicity or alternative approximation arguments.
Orlicz Spaces: In some cases of non-standard growth, Orlicz spaces, which generalize the classical $L^p$ spaces, can provide a suitable framework for analysis. However, their use often introduces technical complications.
3. Challenges and Open Questions:
Sharpness of Exponents: Determining the sharp dependence of the Hölder exponent on the growth and regularity parameters of the integrand becomes even more delicate in the non-quadratic case.
Counterexamples: Constructing counterexamples to demonstrate the optimality of the obtained exponents is crucial but can be highly non-trivial.
Anisotropic Growth: Some integrands exhibit different growth rates in different directions (anisotropic growth). This requires anisotropic function spaces and tailored regularity techniques.
In summary, extending the results to more general growth conditions necessitates a careful adaptation of existing techniques and potentially the development of new tools. The sharpness of the Hölder exponent and the construction of counterexamples pose significant challenges.

Yes, besides Morrey-Hölder conditions, there are alternative structural assumptions on the zero-order terms that can still lead to partial regularity with a quantifiable Hölder exponent. Here are a few examples:
1. VMO (Vanishing Mean Oscillation) Conditions:
Instead of requiring Hölder continuity, one can impose a weaker VMO condition on the zero-order term. This means that the mean oscillation of the function over balls shrinks to zero as the radius of the ball goes to zero.
VMO functions are closer to $L^\infty$ functions than to general $L^p$ functions, and this improved regularity can still be sufficient to obtain partial regularity.
2. Lorentz Space Conditions:
Lorentz spaces provide a finer scale of function spaces between $L^p$ spaces. Imposing conditions on the zero-order term in terms of suitable Lorentz spaces can lead to partial regularity results.
These spaces are particularly useful when dealing with functions that have singularities, as they can capture the decay of the function near singularities more precisely than $L^p$ spaces.
3. Conditions Involving Oscillation or Maximal Functions:
One can consider conditions that directly control the oscillation of the zero-order term over balls, such as bounds on its sharp maximal function or its John-Nirenberg space norm.
These conditions can be weaker than Morrey-Hölder conditions but still provide enough control to establish partial regularity.
4. Structure Conditions Related to Hardy Spaces:
In some cases, imposing conditions on the zero-order term that relate it to functions in Hardy spaces (spaces of functions with good cancellation properties) can be fruitful.
This approach is particularly relevant when dealing with problems involving singular integrals or operators with limited smoothing properties.
Challenges and Considerations:
Sharpness of Exponents: Determining the optimal Hölder exponent under these alternative assumptions might require different techniques and could be more challenging.
Verifying Assumptions: Checking whether a given zero-order term satisfies these alternative conditions might be more involved than verifying Morrey-Hölder conditions.
Connection to Applications: The choice of appropriate structural assumptions is often guided by the specific applications in mind. Some conditions might be more natural or easier to verify in certain physical or geometric contexts.
In conclusion, while Morrey-Hölder conditions provide a robust framework for establishing partial regularity, exploring alternative structural assumptions on the zero-order terms can lead to new insights and potentially weaker conditions that still guarantee regularity. The choice of suitable assumptions depends on the specific problem and the desired level of precision in quantifying the Hölder exponent.

The refined Massari-type regularity, particularly achieving the optimal Hölder exponent, has significant potential implications for understanding the behavior of singularities in geometric flows and free boundary problems. Here's an elaboration:
1. Geometric Flows:
Mean Curvature Flow: The Massari functional is closely related to the area functional, and its minimizers can be viewed as stationary points for the mean curvature flow. The improved regularity results provide better control on the evolution of surfaces under this flow, especially near singularities.
Prescribed Curvature Flows: More generally, for flows where the velocity of the surface depends on its curvature (e.g., Gauss curvature flow), the refined regularity estimates can help analyze the formation and structure of singularities.
Singularity Analysis: The optimal Hölder exponent provides a finer understanding of the geometric nature of singularities. It can help classify different types of singularities and potentially lead to insights into singularity resolution or the long-time behavior of the flow.
2. Free Boundary Problems:
Regularity of Interfaces: Many free boundary problems involve interfaces that separate different phases or regions. The refined Massari-type regularity can be applied to study the regularity of these interfaces, especially when the curvature of the interface plays a role in the problem.
Obstacle Problems: In obstacle problems, the solution is constrained to lie above a given obstacle. The contact set between the solution and the obstacle forms a free boundary. The improved regularity results can provide insights into the smoothness of this free boundary.
Optimal Control Problems: Some optimal control problems lead to free boundary problems, where the control action determines the location of the free boundary. The refined regularity estimates can be used to analyze the regularity of the optimal control and the corresponding free boundary.
3. Specific Examples and Applications:
Fluid Dynamics: Free boundary problems arise in fluid dynamics, for example, in modeling the interface between two fluids or the shape of a droplet. The refined regularity results can help understand the behavior of these interfaces.
Material Science: In material science, free boundary problems model phenomena like phase transitions or the growth of crystals. The improved regularity estimates can provide insights into the microstructure of materials.
Image Processing: Some image segmentation techniques can be formulated as free boundary problems. The refined regularity results can be used to analyze the smoothness of the segmented boundaries.
Overall Impact:
The refined Massari-type regularity results provide a powerful tool for analyzing the regularity and behavior of singularities in a wide range of geometric flows and free boundary problems. They offer a deeper understanding of the geometric and analytical properties of these problems and can lead to advancements in various fields, including fluid dynamics, material science, and image processing.

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