In this paper linear stochastic transport and continuity equations with drift in critical L p spaces are considered. In this situation noise prevents shocks for the transport equation and singularities in the density for the continuity equation, starting from smooth initial conditions. Specifically, we first prove a result of Sobolev regularity of solutions, which is false for the corresponding deterministic equation. The technique needed to reach the critical case is new and based on parabolic equations satisfied by moments of first derivatives of the solution, opposite to previous works based on stochastic flows. The approach extends to higher order derivatives under more regularity of the drift term. By a duality approach, these regularity results are then applied to prove uniqueness of weak solutions to linear stochastic continuity and transport equations and certain well-posedness results for the associated stochastic differential equation (sDE) (roughly speaking, existence and uniqueness of flows and their C α regularity, strong uniqueness for the sDE when the initial datum has diffuse law). Finally, we show two types of examples: on the one hand, we present well-posed sDEs, when the corresponding ODEs are ill-posed, and on the other hand, we give a counterexample in the supercritical case.We first choose α = 2 such that the stochastic part λ α/2−1 ∇u λ (t, x) • W λ (t) is comparable to the derivative in time ∂ t u λ . Notice that this is the parabolic scaling, although sTE is not parabolic (but as we will see below, a basic idea of our approach is that certain expected values of the solution satisfy parabolic equations for which the above scaling is the relevant one). Next we require that, for small λ, the rescaled drift b λ becomes small (or at least controlled) in some suitable norm (in our case, L q (0, T ; L p (R d , R d ))). It is easy to see that b λ L q (0,T /λ 2 ;L p ) = λ 1−(2/q+d/p) b L q (0,T ;L p ) (here, the exponent d comes from rescaling in space and the exponent 2 from rescaling in time and the choice α = 2). In conclusion, we find that• if LPS holds with strict inequality, then b λ L q (0,T /λ 2 ;L p ) → 0 as λ → 0: the stochastic term dominates and we expect a regularizing effect (subcritical case);• if LPS holds with equality, then b λ L q (0,T /λ α ;L p ) = b L q (0,T ;L p ) remains constant: the deterministic drift and the stochastic forcing are comparable (critical case).This intuitively explains why the analysis of the critical case is more difficult. Notice that, if LPS does not hold, then we expect the drift to dominate, so that a general result for regularization by noise is probably false. In this sense, LPS condition should be regarded as an optimal condition for expecting regularity of solutions.

We consider nonuniformly elliptic variational problems and give optimal conditions guaranteeing the local Lipschitz regularity of solutions in terms of the regularity of the given data. The analysis catches the main model cases in the literature. Integrals with fast, exponential‐type growth conditions as well as integrals with unbalanced polynomial growth conditions are covered. Our criteria involve natural limiting function spaces and reproduce, in this very general context, the classical and optimal ones known in the linear case for the Poisson equation. Moreover, we provide new and natural growth a priori estimates whose validity was an open problem. Finally, we find new results also in the classical uniformly elliptic case. Beyond the specific results, the paper proposes a new approach to nonuniform ellipticity that, in a sense, allows us to reduce nonuniform elliptic problems to uniformly elliptic ones via potential theoretic arguments that are for the first time applied in this setting. © 2019 Wiley Periodicals, Inc.

We investigate the Dirichlet problem for multidimensional variational integrals with linear growth which is formulated in a generalized way in the space of functions of bounded variation. We prove uniqueness of minimizers up to additive constants and deduce additional assertions about these constants and the possible (non-)attainment of the boundary values. Moreover, we provide several related examples. In the case of the model integral Ð W ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 þ j'wj 2 q dx for w : R n I W ! R N our results extend classical results from the scalar case N ¼ 1-where the problem coincides with the non-parametric least area problem-to the general vectorial setting N A N.

We investigate the properties of certain elliptic systems leading, a priori, to solutions that belong to the space of Radon measures. We show that if the problem is equipped with a so-called asymptotic Uhlenbeck structure, then the solution can in fact be understood as a standard weak solution, with one proviso: analogously as in the case of minimal surface equations, the attainment of the boundary value is penalized by a measure supported on (a subset of) the boundary, which, for the class of problems under consideration here, is the part of the boundary where a Neumann boundary condition is imposed.1.2. Formulation of the problems. We first consider the following problem: for a bounded, connected, Lipschitz domain Ω ⊂ R d , d ≥ 2, with Dirichlet boundary Γ D and Neumann boundary Γ N , which are relatively open subsets of ∂Ω such that Γ D ∩ Γ N = ∅ and Γ D ∪ Γ N = ∂Ω, a given vector field f : Ω → R N , with N ∈ N, a given g : Γ N → R N , a given boundary datum u 0 : Ω → R N , and a given bounded mapping D D D :where n denotes the unit outward normal vector on Γ N . When Γ D = ∅, f and g will be assumed to satisfy a standard compatibility condition (cf. (D3) below).1 a , a > 0.Problem (1.8) is then an almost direct analogue of problem (1.1) with N = d; the only aspect in which the latter model differs from (1.1) (and is therefore considerably more difficult) is that, in contrast with (1.1), one is forced to operate in the space of symmetric matrices and function spaces of symmetric gradients. We refer the interested reader to [17,18,19,10,9] for a detailed overview of limiting strain models, their theoretical justification stemming from implicit constitutive theory, a discussion of their importance in modeling the responses of materials near regions of stress-concentration, where |T T T| is large, and their mathematical analysis (see in particular the survey paper [9] for more details).Analogously to problem (1.1), we adopt the following natural assumptions associated with limiting strain models (see [9]): there exist constants C 0 ≥ 0 and C 1 , C 2 > 0 such that, for all T T T ∈ R d×d sym , ε ε ε *

We prove partial Hölder continuity for the gradient of minimizers u ∈ W 1,p (Ω, R N ), Ω ⊂ R n a bounded domain, of variational integrals of the formwhere f is strictly quasi-convex and satisfies standard continuity and growth conditions, but where h is only a Caratheodory function of subcritical growth. The main focus is set on the presentation of a unified approach for the interior and the boundary estimates (provided that the boundary data are sufficiently regular) for all p ∈ (1, ∞). Furthermore, a corresponding lower order Hölder regularity result for u is given in dimensions n ≤ p + 2 under the stronger assumption that f is strictly convex.Furthermore, it can be verified that the conditions (1.2) above, see [52, Theorem 4.3], imply the strict ellipticity of the matrix D zz f in the sense of Legendre-Hadamard, and therefore we may also assumefor all ξ ∈ R N , η ∈ R n . As observed by Acerbi and Fusco [2], a growth condition for the second derivatives is not needed, and the continuity of D zz f is sufficient to prove a partial regularity result. This latter condition implies in particular that second order derivatives are bounded on compact subsets ofΩ × R N × R nN , i. e. for every M > 0 there exists a constant K M such that there holds