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We are concerned with interior and global gradient estimates for solutions to a class of singular quasilinear elliptic equations with measure data, whose prototype is given by the p-Laplace equation --Δpu = μ with p ∈ (1, 2). The cases when p ∈ (2 -- 1/n, 2) and p ∈ (3n--2/2n--1, 2 -- 1/n] were studied in Duzaar and Mingione [J. Funct. Anal. 259 (2010), 379-418] and Nguyen and Phuc [J. Funct. Anal. 278 (2020), art. 108391] respectively. In this paper, we improve the results of Nguyen and Phuc and address the open case when p ∈ (1, 3n--2/2n--1)]. Interior and global modulus of continuity estimates of the gradients of solutions are also established. [ABSTRACT FROM AUTHOR]

We obtain the gradient estimates of the positive solutions to a nonlinear elliptic equation on an n-dimensional complete Riemannian manifold (M , g) Δ u + a u (ln u) p + b u ln u = 0 , where a ≠ 0 , b are two constants and p = k 1 2 k 2 + 1 ≥ 2 , here k 1 and k 2 are two positive integers. The gradient bound is independent of the bounds of the solution and the Laplacian of the distance function. As the applications of the estimates, we show the Harnack inequality and the upper bound of the solution. [ABSTRACT FROM AUTHOR]

Let (N, g) be a complete noncompact Riemannian manifold with Ricci curvature bounded from below. In this paper, we study the gradient estimates of positive solutions to a class of nonlinear elliptic equations Δ u (x) + a (x) u (x) log u (x) + b (x) u (x) = 0 on N where a(x) is C2-smooth while b(x) is C1 and its parabolic counterparts (Δ − ∂ ∂ t ) u (x , t) + a (x , t) u (x , t) log u (x , t) + b (x , t) u (x , t) = 0 on N × [0, ∞) where a(x, t) and b(x, t) are C2 with respect to x ∊ N while are C1 with respect to the time t. In contrast with lots of similar results, here we do not assume the coefficients of equations are constant, so our results can be viewed as extensions to several classical estimates. [ABSTRACT FROM AUTHOR]

In this article, we present new gradient estimates for positive solutions to a class of non‐linear elliptic equations involving the f‐Laplacian on a smooth metric measure space. The gradient estimates of interest are of Souplet–Zhang and Hamilton types, respectively, and are established under natural lower bounds on the generalised Bakry–Émery Ricci curvature tensor. From these estimates, we derive amongst other things Harnack inequalities and general global constancy and Liouville‐type theorems. The results and approach undertaken here provide a unified treatment and extend and improve various existing results in the literature. Some implications and applications are presented and discussed. [ABSTRACT FROM AUTHOR]

We are concerned with global gradient estimates of very weak solutions for a general class of quasilinear elliptic equations with nonstandard growth. We establish such estimates under basic structure assumptions on the nonlinearity and natural geometric assumption on the complement of the bounded domain. Our approach is based on a generalized Lipschitz truncation alongside a generalized self-improving property of thick sets. This work provides a careful analysis of the thickness described in Adimurthi and Phuc (Calc Var Partial Differ Equ 54:3107–3139, 2015), Harjulehto and Hästö (Z Anal Anwend 38(1):73–96, 2019), Heinonen et al (Potential theory of degenerate elliptic equations, Oxford University Press, Oxford, 1993). [ABSTRACT FROM AUTHOR]

In this paper, we establish local and global elliptic type gradient estimates for a nonlinear parabolic equation on a smooth metric measure space whose underlying metric and potential satisfy a (k , m) -super Perelman–Ricci flow inequality. We discuss a number of applications and implications including curvature free global estimates and some constancy and Liouville type results. [ABSTRACT FROM AUTHOR]

In this paper, we consider gradient estimates for positive solutions to the following equation △V u + aup log u = 0 on complete noncompact Riemannian manifold with k -dimensional Bakry-Émery Ricci curvature bounded from below. Using the Bochner formula and the Cauchy inequality, we obtain upper bounds of |∇u| with respect to the lower bound of the Bakry-Émery Ricci curvature. [ABSTRACT FROM AUTHOR]

We study the global regularity in weighted Lorentz spaces for the gradient of weak solutions to p-Laplace equations in an open-bounded non-smooth domain which satisfies a p-capacity condition. Moreover, the gradient estimate is presented in terms of fractional maximal operators which are related to the Riesz potential and the fractional derivative. Our main concern is to extend the results of Tran and Nguyen [New gradient estimates for solutions to quasi-linear divergence form elliptic equations with general Dirichlet boundary data. J Differ Equ. 2020;268(4):1427–1462] but in the weighted Lorentz spaces associated to the Muckenhoupt weights. The key technique that we use in this paper is the good-λ inequality and the interesting properties of the cut-off fractional maximal operators. [ABSTRACT FROM AUTHOR]

We investigate a general class of the so-called double phase problems on bounded domains. We establish an optimal global Calderón-Zygmund theory for such a non-uniformly elliptic problem under the assumptions that the coefficients have a small BMO semi-norm and the domain is sufficiently flat. Our regularity results not only cover discontinuous coefficients but also non-smooth domains going beyond the Lipschitz category. [ABSTRACT FROM AUTHOR]

We investigate the boundary regularity of minimizers of convex integral functionals with nonstandard p , q -growth and with Uhlenbeck structure. We consider arbitrary convex domains Ω and homogeneous Dirichlet data on some part Γ ⊂ ∂ Ω of the boundary. For the integrand we assume only a non-standard p , q -growth condition. We establish Lipschitz regularity of minimizers up to Γ, provided the gap between the growth exponents p and q is not too large, more precisely if 1 < p ≤ q < p (1 + 2 n). To our knowledge, this is the first boundary regularity result under a non-standard p , q -growth condition. [ABSTRACT FROM AUTHOR]

In high-contrast composite materials, the electric field concentration is a common phenomenon when two inclusions are close to touch. It is important from an engineering point of view to study the dependence of the electric field on the distance between two adjacent inclusions. In this paper, we derive upper and lower bounds of the gradient of solutions to the conductivity problem where two perfectly conducting inclusions are located very close to each other. To be specific, we extend the known results of Bao-Li-Yin (ARMA 2009) in two folds: First, we weaken the smoothness of the inclusions from C 2 , α to C 1 , α. To obtain a pointwise upper bound of the gradient, we follow an iteration technique which is first used to deal with elliptic systems in a narrow domain by Li-Li-Bao-Yin (QAM 2014). However, when the inclusions are of C 1 , α , we can not use W 2 , p estimates for elliptic equations any more. In order to overcome this new difficulty, we take advantage of De Giorgi-Nash estimates and Campanato's approach to apply an adapted version of the iteration technique with respect to the energy. A lower bound in the shortest line between two inclusions is also obtained to show the optimality of the blow-up rate. Second, when two inclusions are only convex but not strictly convex, we prove that blow-up does not occur any more. The establishment of the relationship between the blow-up rate of the gradient and the order of the convexity of the inclusions reveals the mechanism of such concentration phenomenon. [ABSTRACT FROM AUTHOR]

We study regularity for solutions of quasilinear elliptic equations of the form divA(x, u,u) = divF in bounded domains in Rn. The vector field A is assumed to be continuous in u, and its growth in u is like that of the p-Laplace operator. We establish interior gradient estimates in weighted Morrey spaces for weak solutions u to the equation under a small BMO condition in x for A. As a consequence, we obtain that u is in the classical Morrey space Mq, or weighted space Lq w whenever |F|1/(p-1) is respectively in Mq, or Lq w, where q is any number greater than p and w is any weight in the Muckenhoupt class Aq/p. In addition, our twoweight estimate allows the possibility to acquire the regularity for u in a weighted Morrey space that is different from the functional space that the data |F|1/(p-1) belongs to. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present the global estimate for gradient of renormalized solutions to the following quasilinear elliptic problem: − div (A (x , ∇ u)) = μ in Ω , u = 0 on ∂ Ω , in Lorentz–Morrey spaces, where Ω ⊂ ℝ n (n ≥ 2), μ is a finite Radon measure, A is a monotone Carathéodory vector-valued function defined on W 0 1 , p (Ω) and the p -capacity uniform thickness condition is imposed on the complement of our domain Ω. It is remarkable that the local gradient estimates have been proved first by Mingione in [Gradient estimates below the duality exponent, Math. Ann.346 (2010) 571–627] at least for the case 2 ≤ p ≤ n , where the idea for extending such result to global ones was also proposed in the same paper. Later, the global Lorentz–Morrey and Morrey regularities were obtained by Phuc in [Morrey global bounds and quasilinear Riccati type equations below the natural exponent, J. Math. Pures Appl.102 (2014) 99–123] for regular case p > 2 − 1 n . Here in this study, we particularly restrict ourselves to the singular case 3 n − 2 2 n − 1 < p ≤ 2 − 1 n . The results are central to generalize our technique of good- λ type bounds in the previous work [M.-P. Tran, Good- λ type bounds of quasilinear elliptic equations for the singular case, Nonlinear Anal.178 (2019) 266–281], where the local gradient estimates of solution to this type of equation were obtained in the Lorentz spaces. Moreover, the proofs of most results in this paper are formulated globally up to the boundary results. [ABSTRACT FROM AUTHOR]

This paper studies a new gradient regularity in Lorentz spaces for solutions to a class of quasilinear divergence form elliptic equations with nonhomogeneous Dirichlet boundary conditions: { div (A (x , ∇ u)) = div (| F | p − 2 F) in Ω , u = σ on ∂ Ω , where Ω ⊂ R n (n ≥ 2), the nonlinearity A is a monotone Carathéodory vector valued function defined on W 0 1 , p (Ω) for p > 1 and the p -capacity uniform thickness condition is imposed on the complement of our bounded domain Ω. Moreover, for given data F ∈ L p (Ω ; R n) , the problem is set up with general Dirichlet boundary data σ ∈ W 1 , p (Ω). In this paper, the optimal good- λ type bounds technique is applied to prove some results of fractional maximal estimates for gradient of solutions. And the main ingredients are the action of the cut-off fractional maximal functions and some local interior and boundary comparison estimates developed in previous works [45,52,53] and references therein. [ABSTRACT FROM AUTHOR]

We study global gradient estimates for the weak solution to elliptic systems with measurable coefficients from composite materials in a nonsmooth bounded domain. The principal coefficients are assumed to be merely measurable in one variable and have small bounded mean oscillation (BMO) seminorms in the other variables on each subdomain whose boundary satisfies the so‐called δ‐Reifenberg flat condition. Under these assumptions and based on our new geometric observation for disjoint Reifenberg domains in a previous study, we obtain global W1,p estimates. [ABSTRACT FROM AUTHOR]

In this paper, we construct new barriers for the Pucci extremal operators with unbounded RHS. The geometry of these barriers is given by a Harnack inequality up to the boundary type estimate. Under the possession of these barriers, we prove a new quantitative version of the Hopf–Oleĭnik Lemma for quasilinear elliptic equations with g -Laplace type growth. Finally, we prove (sharp) regularity for ω -semiconvex supersolutions for some nonlinear PDEs. These results are new even for second order linear elliptic equations in nondivergence form. Moreover, these estimates extend and improve a classical a priori estimate proven by L. Caffarelli, J.J. Kohn, J. Spruck and L. Nirenberg in [13] in 1985 as well as a more recent result on the C 1 , 1 regularity for convex supersolutions obtained by C. Imbert in [33] in 2006. [ABSTRACT FROM AUTHOR]

We obtain a global generalized weighted Morrey M... estimate for the gradient of the weak solutions to divergence form elliptic equations with measurable coefficients in a nonsmooth bounded domain. The coefficients are assumed to be merely measurable in one variable and to have small BMO semi-norms in the remaining variables, while the boundary of the domain is supposed to be Reifenberg at, which goes beyond the category of domains with Lipschitz continuous boundaries. As consequence of the main result, we derive global gradient estimate for the weak solution in the framework of the generalized weighted Morrey spaces. [ABSTRACT FROM AUTHOR]

This paper concerns the gradient estimates for Neumann problem of a certain Monge-Ampère type equation with a lower order symmetric matrix function in the determinant. Under a one-sided quadratic structure condition on the matrix function, we present two alternative full discussions of the global gradient bound for the elliptic solutions. [ABSTRACT FROM AUTHOR]

We prove a differential Harnack inequality for the solution of the parabolic Allen-Cahn equation $$ \frac{\partial f}{\partial t}=\triangle f-(f^3-f)$$ on a closed n-dimensional manifold. As a corollary, we find a classical Harnack inequality. We also formally compare the standing wave solution to a gradient estimate of Modica from the 1980s for the elliptic equation. [ABSTRACT FROM AUTHOR]

In this paper, the aim of our work is to establish global weighted gradient estimates via fractional maximal functions and the point-wise regularity estimates of Dirichlet problem for divergence elliptic equations of the type div (A (x , ∇ u)) = div (f) in Ω , and u = g on ∂ Ω , that related to Riesz potentials. Here, in our setting, Ω ⊂ R n , n ≥ 2 is a bounded Reifenberg flat domain (that its boundary is sufficiently flat in sense of Reifenberg) and the small-BMO condition (small bounded mean oscillations) is assumed on the nonlinearity A. Further, the emphasis of the paper is the existence of weak solution to a class of quasilinear elliptic equations containing Riesz potential of the gradient term, as an application of the global point-wise bound. And regarding this study, we also analyze the necessary and sufficient conditions that guarantee the existence of solution to such nonlinear elliptic problems. [ABSTRACT FROM AUTHOR]

In this paper, we will use the maximum principle to give a new proof of the gradient estimates for mean curvature equations with some oblique derivative problems. In particular, we shall give a new proof for the capillary problem with zero gravity. [ABSTRACT FROM AUTHOR]

We consider the quasilinear degenerate elliptic equationwhere Δpis thep-Laplace operator,p > 2, λ ≥0 and Ω is a smooth open bounded subset of ℝN(N ≥ 2). Under suitable structure conditions on the functionH, we prove local and global gradient bounds for the solutions. We apply these estimates to study the solvability of the Dirichlet problem, and the existence, uniqueness and asymptotic behavior of maximal solutions blowing up at the boundary. The ergodic limit for those maximal solutions is also studied and the existence and uniqueness of a so-called additive eigenvalue is proved in this context. [ABSTRACT FROM AUTHOR]

We establish a gradient estimate for a very weak solution to a quasilinear elliptic equation with a nonstandard growth condition, which is a natural generalization of the p -Laplace equation. We investigate the maximum extent for the gradient estimate to hold without imposing any regularity assumption on the nonlinearity other than basic structure assumptions. Our results also include a higher integrability result of the gradient and the existence for the very weak solutions to such nonlinear problems. [ABSTRACT FROM AUTHOR]

We consider a class of Dirichlet boundary problems for nonlinear elliptic equations with a first order term. We show how the summability of the gradient of a solution increases when the summability of the datum increases. We also prove comparison principle which gives in turn uniqueness results by strenghtening the assumptions on the operators. [ABSTRACT FROM AUTHOR]

We prove a mean value formula for weak solutions of $$\hbox {div}(|y|^{a}\hbox {grad} u)=0$$ in $${\mathbb {R}}^{n+1}=\{ (x,y):\ x\in {\mathbb {R}}^{n},\ y\in {\mathbb {R}} \}$$ , $$-1

Optimal regularity estimates are established for the gradient of solutions to non-uniformly elliptic equations of Orlicz double phase with variable exponents type in divergence form under sharp conditions on such highly nonlinear operators for the Calderón–Zygmund theory. [ABSTRACT FROM AUTHOR]

This is a survey of some recent contributions by the authors on global integrability properties of the gradient of solutions to boundary value problems for nonlinear elliptic equations in divergence form. Minimal assumptions on the regularity of the ground domain and of the prescribed data are pursued. [ABSTRACT FROM AUTHOR]

We investigate the growth of entire positive functions u(x) and their gradients Du in Sobolev spaces when a polynomial growth is assumed for their image Lu through a linear second-order uniform elliptic operator L. In particular, under suitable assumptions on the coefficients, we show that if Lu is bounded, then u(x) may grow at most quadratically at infinity. We also discuss, by counterexamples, the optimality of the assumptions and extend the results to viscosity solutions of fully nonlinear equations. [ABSTRACT FROM AUTHOR]

This paper continues the development of regularity results for quasilinear elliptic equations − div (A (x , ∇ u)) = μ in Ω , and u = 0 on ∂ Ω , in Lorentz and Lorentz–Morrey spaces, where Ω ⊂ R n (n ≥ 2); A is a monotone Carathéodory vector valued operator acting between W 0 1 , p (Ω) and its dual W − 1 , p ′ (Ω) ; and μ is a datum in some Lebesgue space L m (Ω) , for m < p ′ . It emphasizes that in this paper, we restrict our study to the case of 'very singular' when 1 < p ≤ 3 n − 2 2 n − 1 , and under mild assumption that the p -capacity uniform thickness condition is imposed on the complement of domain Ω. There are two main results obtained in our study pertaining to the global gradient estimates of solutions in Lorentz and Lorentz–Morrey spaces involving the use of maximal and fractional maximal operators. The idea for writing this working paper comes directly from the recent results by others in the same research topic, where global estimates for gradient of solutions for the 'very singular' case still remains a challenge, specifically related to Lorentz and Lorentz–Morrey spaces. [ABSTRACT FROM AUTHOR]

In this paper, we prove global gradient estimates for solutions to linear elliptic and parabolic equations. For a sufficiently smooth bounded convex domain Ω ⊂ R N , we show that a solution ϕ ∈ W 0 1 , ∞ (Ω ; R) to an appropriate elliptic equation L ϕ = F , with F ∈ L ∞ (Ω ; R) , satisfies | ∇ ϕ | ∞ ≤ C | F | ∞ , with a positive constant C = exp ⁡ (C (L) diam (Ω)). We also obtain similar estimates in the parabolic setting. The proof of these exponential bounds relies on global gradient estimates inspired by a series of papers by Ben Andrews and Julie Clutterbuck. This work is motivated by a dual version of the Landis conjecture. [ABSTRACT FROM AUTHOR]

We prove Calderón-Zygmund type estimates for distributional solutions to non-uniformly elliptic equations of generalized double phase type in divergence form. In particular, we provide sharp conditions on the nonlinear operators to establish the Calderón-Zygmund type estimates. [ABSTRACT FROM AUTHOR]

This paper proves local gradient estimates on positive solutions to the following nonlinear elliptic equation Δ f u + a u (log u) α = 0 , where a and α are real constants, on complete weighted manifolds with Bakry–Émery Ricci tensor bounded from below. For applications, global estimates and some Liouville type theorems are derived. [ABSTRACT FROM AUTHOR]