A. A. Arkhipova

On the regularity of solutions of boundary-value problem for quasilinear elliptic systems with quadratic nonlinearity

The partial regularity up to the boundary of a domain is established for a solution u ∈ H1 (Ω) ∩ L∞ (Ω) to the boundary-value problem for a second-order elliptic system with strong nonlinearity in the case of dimension n≥3. Bibliography: 12 titles.

Regularity of the solutions of diagonal elliptic systems under convex constraints on the boundary of the domain

One investigates the smoothness of the solutions of variational inequalities, connected with second-order linear diagonal elliptic systems under convex constraints on the solution at the boundary of the domain. One establishes the Holder continuity of the first derivatives of the solutions up to the boundary of the domain.

On a partial regularity up to the boundary of weak solutions to quasilinear parabolic systems with quadratic growth

Quasilinear nondiagonal parabolic systems with quadratic growth in the gradient in a parabolic cylinder Q are considered. Under Dirichlet and Neumann boundary conditions, a partial Hölder continuity of solutions u∈W2t,1 (Q)∼L∞ (Q) up to the lateral surface of Q is proved.The Hausdorff dimension of a singular set is estimated. In the proof, we get rid of the maximum principle theorem for respective model linear problems. Bibliography: 21 titles.

Limit smoothness of the solutions of variational inequalities under convex constraints on the boundary of the domain

Variational inequalities, connected with quasilinear elliptic systems with a diagonal principal part and a quadratic growth along the gradient, are considered. On the boundary of the domain convex constraints are imposed on the solution. The Hölder continuity of the solution up to the boundary of the domain is proved.

Regularity results for quasilinear elliptic systems with nonlinear boundary conditions

It is proved that a solution of the boundary-value problem for a second-order quasilinear system with controlled order of nonlinearity is partially smooth all the way to the boundary of a domain. The boundary condition is imposed by means of a second-order nonlinear operator which can be regarded as a generalization of the “directional derivative” to the case of quasilinear systems. Bibliography: 6 titles.

Partial Regularity up to the Boundary for Weak Solutions of Elliptic Systems with Nonlinearity q Greater Than Two

Nonlinear elliptic systems with q-growth are considered. It is assumed that additional nonlinear terms of the systems have q-growth in the gradient, q < 2. For Dirichlet and Neumann boundary-value problems we study the regularity of weak bounded solutions in the vicinity of the boundary. In the case of small dimensions (n ≤ q + 2), the Hölder continuity or partial Hölder continuity up to the boundary is proved for the solutions considered. In the previous article, the author studied the same problem for q = 2. Bibliography: 12 titles.

Modifications of the Gehring lemma appearing in the study of parabolic initial-boundary-value problems

Some modifications of the Gehring lemma are required in the study of solutions to parabolic initialboundary-value problems, The Gehring lemma assert that if a function satisfies the reverse Hölder inequalitites in a cube, then the integrability degree of this function in Q increases in this cube. Earlier, the author formulated some generalizations of the Gehring lemma and used them in the study of parabolic quasilinear systems with controlled nonlinearity orders. In this paper, the proof of these generalizations are given. On the basis of the modification of the Gehring lemma proposed by the author, the theorem on the reverse Hölder inequalities is formulated in a form convenient for obtaining Lp-estimates for the derivatives of solutions to parabolic problems. An application of this theorem is also demonstrated. Bibliography: 19 titles.

Global solvability of the Cauchy-Dirichlet problem for nondiagonal parabolic systems with variational structure in the case of two spatial variables

The Cauchy-Dirichlet problem for quasilinear parabolic systems of second-order equations is considered in the case of two spatial variables. Under the condition that the corresponding elliptic operator has variational structure, the global in time solvability is established. The solution is smooth almost everywhere and the number of singular points is finite. Sufficient conditions that guarantee the absence of singular points are given. Bibliography: 23 titles.

On the regularity of the oblique derivative problem for quasilinear elliptic systems

This note continues the authors investigations of the regularity problem for quasilinear elliptic systems with nonlinear boundary conditions. Partial regularity of weak solutions of the oblique derivative problem is proved here. Bibliography: 7 titles.

Regularity of Solutions to a Diffraction-Type Problem for Nondiagonal Linear Elliptic Systems in the Campanato Space

We study a two-phase difraction-type problem for nondiagonal linear elliptic systems of equations. We study the regularity of a weak solution to the problem in the Campanato spaces. In particular, we prove the smoothness of solutions in a neighborhood of the interface surface separating media. For model systems with constant coefficients we derive Campanato-type integral estimates. Bibliography: 13 titles.