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Motivated by a challenging expectation of Rivière [24] , in the recent interesting work [5] , de Longueville and Gastel proposed the following geometrical even order elliptic system Δ m u = ∑ l = 0 m − 1 Δ l 〈 V l , d u 〉 + ∑ l = 0 m − 2 Δ l δ (w l d u) in B 2 m which includes polyharmonic mappings as special cases. Under minimal regularity assumptions on the coefficient functions and an additional algebraic antisymmetry assumption on the first order potential, they successfully established a conservation law for this system, from which everywhere continuity of weak solutions follows. This beautiful result amounts to a significant advance in the expectation of Rivière. In this paper, we seek for the optimal interior regularity of the above system, aiming at a more complete solution to the aforementioned expectation of Rivière. Combining their conservation law and some new ideas together, we obtain optimal Hölder continuity and sharp L p regularity theory, similar to that of Sharp and Topping [27] , for weak solutions to a related inhomogeneous system. Our results can be applied to study heat flow and bubbling analysis for polyharmonic mappings. [ABSTRACT FROM AUTHOR]

Let M be a C 2 -smooth Riemannian manifold with boundary and N a complete C 2 -smooth Riemannian manifold. We show that each stationary p -harmonic mapping u : M → N , whose image lies in a compact subset of N , is locally C 1 , α for some α ∈ (0 , 1) , provided that N is simply connected and has non-positive sectional curvature. We also prove similar results for minimizing p -harmonic mappings with image being contained in a regular geodesic ball. Moreover, when M has non-negative Ricci curvature and N is simply connected with non-positive sectional curvature, we deduce a gradient estimate for C 1 -smooth weakly p -harmonic mappings from which follows a Liouville-type theorem in the same setting. [ABSTRACT FROM AUTHOR]

In this article our main concern is to prove the quantitative unique estimates for the p -Laplace equation, 1 < p < ∞ , with a locally Lipschitz drift in the plane. To be more precise, let u ∈ W l o c 1 , p ( R 2 ) be a nontrivial weak solution to div ( | ∇ u | p − 2 ∇ u ) + W ⋅ ( | ∇ u | p − 2 ∇ u ) = 0 in R 2 , where W is a locally Lipschitz real vector satisfying ‖ W ‖ L q ( R 2 ) ≤ M ̃ for q ≥ max { p , 2 } . Assume that u satisfies certain a priori assumption at 0. For q > max { p , 2 } or q = p > 2 , if ‖ u ‖ L ∞ ( R 2 ) ≤ C 0 , then u satisfies the following asymptotic estimates at R ≫ 1 inf | z 0 | = R sup | z − z 0 | < 1 | u ( z ) | ≥ e − C R 1 − 2 q log R , where C > 0 depends only on p , q , M ̃ and C 0 . When q = max { p , 2 } and p ∈ ( 1 , 2 ] , if | u ( z ) | ≤ | z | m for | z | > 1 with some m > 0 , then we have inf | z 0 | = R sup | z − z 0 | < 1 | u ( z ) | ≥ C 1 e − C 2 ( log R ) 2 , where C 1 > 0 depends only on m , p and C 2 > 0 depends on m , p , M ̃ . As an immediate consequence, we obtain the strong unique continuation principle (SUCP) for nontrivial solutions of this equation. We also prove the SUCP for the weighted p -Laplace equation with a locally positive locally Lipschitz weight. [ABSTRACT FROM AUTHOR]

We study L ∞ -variational problems associated to measurable Finsler structures in Euclidean spaces. We obtain existence and uniqueness results for the absolute minimizers. [ABSTRACT FROM AUTHOR]

In this short note, we prove interior regularity of weak solutions to a class of linear elliptic system with quadratic Jacobian structures in higher dimensions, extending the result of Hajlasz, Strzelecki and Zhong (2008) in dimension two. [ABSTRACT FROM AUTHOR]