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Mikhail Surnachev

Publications and source records attributed to Mikhail Surnachev.

9 recordsLinked to original sources
arXiv2026

Gradient estimates for $p\left(\cdot\right)$-harmonic differential forms

In this paper, we establish gradient bounds for $p(\cdot)$-harmonic differential forms subject to a Coulomb-type gauge condition. For variable exponents satisfying the log-H\"older continuity assumption, we derive higher integrability estimates of Meyers type, ensuring improved regularity beyond the natural energy space. Furthermore, under the stronger assumption of H\"older continuity of the exponent function, we prove that the gradient of solutions exhibits H\"older continuity. These results extend classical regularity theory for constant-exponent $p$-harmonic systems to the variable-exponent setting, which is essential for modeling nonhomogeneous and anisotropic media.

math.AP↗
arXiv2025

Hodge Decomposition and Potentials in Variable Exponent Lebesgue and Sobolev Spaces

The objective of this work is to establish a systematic study of boundary value problems within the framework of differential forms and variable exponent spaces. Specifically, we investigate the Hodge Laplacian and related first order systems like the div-curl systems, Hodge-Dirac systems, and Bogovskii-type problems in the context of variable exponent spaces. Our approach yields both existence theorems and elliptic estimates. These estimates provide key results such as the Hodge decomposition theorem, Gaffney inequality, and gauge fixing. These findings are crucial for advancing the nonlinear theory related to problems involving differential forms.

math.AP↗
arXiv2024

On the stability of finite-volume schemes on non-uniform meshes

In this paper, we study the L2 stability of high-order finite-volume schemes for the 1D transport equation on non-uniform meshes. We consider the case when a small periodic perturbation is applied to a uniform mesh. For this case, we establish a sufficient stability condition. This allows to prove the (p+1)-th order convergence of finite-volume schemes based on p-th order polynomials.

math.NA↗
arXiv2024

On the order of accuracy of finite-volume schemes on unstructured meshes

We consider finite-volume schemes for linear hyperbolic systems with constant coefficients on unstructured meshes. Under the stability assumption, they exhibit the convergence rate between $p$ and $p+1$ where $p$ is the order of the truncation error. Our goal is to explain this effect. The central point of our study is that the truncation error on $(p+1)$-th order polynomials has zero average over the mesh period. This condition is verified for schemes with a polynomial reconstruction, multislope finite-volume methods, 1-exact edge-based schemes, and the flux correction method. We prove that this condition is necessary and, under additional assumptions, sufficient for the $(p+1)$-th order convergence. Furthermore, we apply the multislope method to a high-Reynolds number flow and explain its accuracy.

math.NA↗
arXiv2023

The Lavrentiev phenomenon in calculus of variations with differential forms

In this article we study convex non-autonomous variational problems with differential forms and corresponding function spaces. We introduce a general framework for constructing counterexamples to the Lavrentiev gap, which we apply to several models, including the double phase, borderline case of double phase potential, and variable exponent. The results for the borderline case of double phase potential provide new insights even for the scalar case, i.e., variational problems with $0$-forms.

math.AP↗
arXiv2020

Lavrentiev gap for some classes of generalized Orlicz functions

In the present paper we find optimal conditions separating the regular case from the one with Lavrentiev gap for the borderline case of double phase potencial and related general classes of integrands. We present new results on density of smooth functions.

math.AP↗
arXiv2019

New Examples on Lavrentiev Gap Using Fractals

Zhikov showed 1986 with his famous checkerboard example that functionals with variable exponents can have a Lavrentiev gap. For this example it was crucial that the exponent had a saddle point whose value was exactly the dimension. In 1997 he extended this example to the setting of the double phase potential. Again it was important that the exponents crosses the dimensional threshold. Therefore, it was conjectured that the dimensional threshold plays an important role for the Lavrentiev gap. We show that this is not the case. Using fractals we present new examples for the Lavrentiev gap and non-density of smooth functions. We apply our method to the setting of variable exponents, the double phase potential and weighted p-energy.

math.AP↗