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Science China Mathematics 2007年第1期50卷 105-115页

作者： Shu-hong CHEN Zhong TAN School of Mathematical Science Xiamen UniversityXiamen 361005China

In this paper, we are concerned with the partial regularity for the weak solutions of energy minimizing p-harmonic maps under the controllable growth condition. We get the interior partial regularity by the p-harmonic approximation method together with the technique used to get the decay estimation on some Degenerate elliptic equations and the obstacle problem by Tan and Yan. In particular, we directly get the optimal regularity.

关键词： p-harmonic approximation controllable growth condition regularity 35j70 35j60 35D10 35B65

Oblique derivative problem for general Chaplygin-Rassias equations

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Science China Mathematics 2008年第1期51卷 5-36页

作者： WEN GuoChun LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China LMAM School of Mathematical Sciences Peking University Beijing China

The present paper deals with the oblique derivative problem for general second order equations of mixed (elliptic-hyperbolic) type with the nonsmooth parabolic degenerate line $$K_1 (y)u_{xx} + \left| {K_2 (x)} \right|u_{yy} + a(x,y)u_x + b(x,y)u_y + c(x,y)u = - d(x,y)$$ in any plane domain D with the boundary ?D=Γ ∪ L 1 ∪ L 2 ∪ L 3 ∪ L 4, where Γ(? {y > 0}) ∈ C μ 2 (0 < μ < 1) is a curve with the end points z = ?1, 1. L 1, L 2, L 3, L 4 are four characteristics with the slopes ?H 2(x)/H 1(y), H 2(x)/H 1(y),?H 2(x)/H 1(y),H 2(x)/H 1(y) (H 1(y) = √|K 1(y)|, H 2(x) = √|K 2(x)| in {y < 0}) passing through the points z = x + iy = ?1, 0, 0, 1 respectively. And the boundary condition possesses the form $$\frac{1}{2}\frac{{\partial u}}{{\partial \nu }} = \frac{1}{{H(x,y)}}\operatorname{Re} \left[ {\overline {\lambda (z)} u_{\tilde z} } \right] = r(z), z \in \Gamma \cup L_1 \cup L_4 , \operatorname{Im} \left[ {\overline {\lambda (z)} u_{\tilde z} } \right]\left| {_{z = z_l } } \right. = b_l ,l = 1,2, u( - 1) = b_0 ,u(1) = b_3 ,$$ in which z 1, z 2 are the intersection points of L 1, L 2, L 3, L 4 respectively. The above equations can be called the general Chaplygin-Rassias equations, which include the Chaplygin-Rassias equations $$K_1 (y)(M_2 (x)u_x )_x + M_1 (x)(K_2 (y)u_y )_y + r(x,y)u = f(x,y), in D$$ as their special case. The above boundary value problem includes the Tricomi problem of the Chaplygin equation: K(y)u xx+u yy = 0 with the boundary condition u(z) = ?(z) on Γ ∪ L 1 ∪ L 4 as a special case. Firstly some estimates and the existence of solutions of the corresponding boundary value problems for the degenerate elliptic and hyperbolic equations of second order are discussed. Secondly, the solvability of the Tricomi problem, the oblique derivative problem and Frankl problem for the general Chaplygin-Rassias equations are proved. The used method in this paper is different from those in other papers, because the new notations W(z) = W(x + iy) = $u_{\tilde z} $ = [H

关键词： oblique derivative problem equations of mixed type nonsmooth degenerate line 35j70 35L80 35N99

The prescribed p-mean curvature equation of low regularity in the Heisenberg group

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Science China Mathematics 2009年第12期52卷 2604-2609页

作者： CHENG jih-Hsin Institute of Mathematics Academia Sinica

This work reports on the author's recent study about regularity and the singular set of a C 1 smooth surface with prescribed p (or H)-mean curvature in the 3-dimensional Heisenberg *** a differential equation,this is a degenerate hyperbolic and elliptic PDE of second order,arising from the study of CR *** only the p-mean curvature H ∈ C 0,it is shown that any characteristic curve is C 2 smooth and its (line) curvature *** introducing special coordinates and invoking the jump formulas along characteristic curves,it is proved that the Legendrian (horizontal) normal gains one more *** the seed curves are C 2 *** work also obtains the uniqueness of characteristic and seed curves passing through a common point under some mild conditions,*** an on-going project,it is shown that the p-area element is in fact C 2 smooth along any characteristic curve and satisfies a certain ordinary differential equation of second ***,this ODE is analyzed to study the singular set.

关键词： Heisenberg group p-minimal surface Bernstein-type theorem 35L80 35j70 32V20 53A10 49Q10

Carleman estimates and unique continuation property for the anisotropic differential-operator equations

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Science China Mathematics 2008年第7期51卷 1215-1231页

作者： Veli B SHAKHMUROV Department of Mathematics Okan UniversityAkfirat34959 TuzlaIstanbulTurkey

The unique continuation theorems for the anisotropic partial differential-operator equations with variable coefficients in Banach-valued L p -spaces are studied. To obtain the uniform maximal regularity and the Carleman type estimates for parameter depended differential-operator equations, the sufficient conditions are founded. By using these facts, the unique continuation properties are established. In the application part, the unique continuation properties and Carleman estimates for finite or infinite systems of quasielliptic partial differential equations are studied.

关键词： Carleman estimates unique continuation embedding theorems Banach-valued function spaces differential operator equations maximal L p -regularity operator-valued Fourier multipliers interpolation of Banach spaces 34G10 35j25 35j70