A boundary regularity result for minimizers of variational integrals with nonstandard growth

We prove global Lipschitz regularity for a wide class of convex variational integrals among all functions in $W^{1,1}$ with prescribed (sufficiently regular) boundary values, which are not assumed to satisfy any geometrical constraint (as for example bounded slope condition). Furthermore, we do not assume any restrictive assumption on the geometry of the domain and the result is valid for all sufficiently smooth domains. The result is achieved with a suitable approximation of the functional together with a new construction of appropriate barrier functions.Comment: arXiv admin note: text overlap with arXiv:1310.6845 by other author

Internal Schauder estimates for H\"ormander type equations with Dini continuous source

We study the regularity properties of a general second order H\"ormander operator with Dini continous coefficients $a_{ij}$. Precisely if $X_0, X_1,\cdots X_m$ are smooth self adjoint vector fields satisfying the H\"ormander condition, we consider the linear operator in $\mathbb{R}^{N}$, with $N>m+1$: \begin{equation*} \mathcal{L} u := \sum_{i, j= 1}^{m} a_{ij} X_{i}X_{j} u - X_0 u. \end{equation*} The vector field $X_0$ plays a role similar to the time derivative in a parabolic problem so that it is a vector of degree two. We prove that, if $f$ is a Dini continuous function, then the second order derivatives of the solution $u$ to the equation $\mathcal{L} u = f$ are Dini continuous functions as well. A key step in our proof is a Taylor formula in this anisotropic setting, that we establish under minimal regularity assumptions.Comment: arXiv admin note: substantial text overlap with arXiv:2102.1038

A VERSION OF THE {H}OPF-{L}AX FORMULA IN THE {H} EISENBERGGROUP

We consider Hamilton-Jacobi equations in the Heisenberg group . We establish uniqueness of bounded viscosity solutions with continuous initial data . When the hamiltonian H is radial, convex and superlinear the solution is given by the Hopf-Lax formula where the Lagrangian L is the horizontal Legendre transform of H lifted to the algebra by requiring it to be radial with respect to the Carnot-Carathéodory metric

Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients

We study the regularity properties of the second order linear degenerate parabolic operators. We prove that, if the operator Lsatisfies Hörmander’s hypoellipticity condition, and f is a Dini continuous function, then the second order derivatives of the solution u to the equation L u = f are Dini continuous functions as well. We also consider the case of Dini continuous coefficients of the secondo order derivatives. A key step in our proof is a Taylor formula for classical solutions to L u = f that we establish under minimal regularity assumptions on u

Interior and boundary continuity of the solution of the singular equation (beta(u))t=Lu(beta(u))_ t= Lu

We extendsome results of DiBenedetto and Vespri (Arch. Rational Mech. Anal 132(3) (1995) 247) proving the interior and boundary continuity of bounded solutions of the singular equation

Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients

We study the regularity properties of the second order linear operator in $R^{N+1}$: egin{equation*} L u := sum_{j,k= 1}^{m} a_{jk}partial_{x_j x_k}^2 u + sum_{j,k= 1}^{N} b_{jk}x_k partial_{x_j} u - partial_t u, end{equation*} where $A = left( a_{jk} ight)_{j,k= 1, dots, m}, B= left( b_{jk} ight)_{j,k= 1, dots, N}$ are real valued matrices with constant coefficients, with $A$ symmetric and strictly positive. We prove that, if the operator $L$ satisfies H"ormander's hypoellipticity condition, and $f$ is a Dini continuous function, then the second order derivatives of the solution $u$ to the equation $L u = f$ are Dini continuous functions as well. We also consider the case of Dini continuous coefficients $a_{jk}$'s. A key step in our proof is a Taylor formula for classical solutions to $L u = f$ that we establish under minimal regularity assumptions on $u$

Integral representation and Γ\Gamma-convergence for free-discontinuity problems with p(⋅)p(\cdot)-growth

An integral representation result for free-discontinuity energies defined on the space $GSBV^{p(\cdot)}$ of generalized special functions of bounded variation with variable exponent is proved, under the assumption of log-H\"older continuity for the variable exponent $p(x)$. Our analysis is based on a variable exponent version of the global method for relaxation devised in Bouchitt\`e, Fonseca, Leoni and Mascarenhas '98 for a constant exponent. We prove $\Gamma$-convergence of sequences of energies of the same type, we identify the limit integrands in terms of asymptotic cell formulas and prove a non-interaction property between bulk and surface contributions

Interior and Boundary Continuity of the Solution of the Singular Equation (beta(u))t=Lu(beta(u))_t=Lu

We deal with the singular parabolic equation $$(beta (u))_{t}={cal L} u,$$ of the kind arising in the modelling of phase transition for fluids and prove interior and boundary continuity of the weak solutions. With respect to previous results due to DiBenedetto and Vespri where ${cal L}$ is simply the Laplace operator, here we consider a more general situation, where ${cal L}$ is a second order elliptic operator with bounded and measurable coefficients that depend both on space and time in a proper way. The main focus is on the interior behaviour, where special care has to be used in order to deal with the lack of radial simmetry of ${cal L}$. We first develop the case of time - independent coefficients and then consider the general case by a perturbation argument. With regards to the boundary behaviour, we deal with homogeneous Dirichlet conditions. This is simply a first step towards a comprehensive study of general Dirichlet and Neumann conditions for this kind of equations