The initial-irregular oblique derivative boundary value problems for nonlinear and nondivergence parabolic systems of second order equations in multiply connected domains are dealt with where coefficients of systems o...The initial-irregular oblique derivative boundary value problems for nonlinear and nondivergence parabolic systems of second order equations in multiply connected domains are dealt with where coefficients of systems of equations are meaurable. The uniqueness theorem of solutions for the above problems and some a priori estimates of solutions for the problems are given. And by using the above estimates of solutions and the Leray-Schauder theorem, the existence of solutions of the initial-boundary value problems is proved. The results are generalizations of corresponding theorems in literature.展开更多
The present article deals with oblique derivative problems for some nonlinear mixed equations with parabolic degeneracy, which include the Tricomi problem as a special case. First, the formulation of the problems for ...The present article deals with oblique derivative problems for some nonlinear mixed equations with parabolic degeneracy, which include the Tricomi problem as a special case. First, the formulation of the problems for the equations is given; next, the representation and estimates of solutions for the above problems are obtained; finally, the existence of solutions for the problems is proved by the successive iteration and the compactness principle of solutions of the problems. In this article, the author uses the complex method, namely, the complex functions in the elliptic domain and the hyperbolic complex functions in hyperbolic domain are used.展开更多
This paper deals with some initial-oblique derivative boundary value problems for nonlinear nondivergent parabolic systems of several second order equations with measurable coefficients in multiply connected domains. ...This paper deals with some initial-oblique derivative boundary value problems for nonlinear nondivergent parabolic systems of several second order equations with measurable coefficients in multiply connected domains. Firstly, a priori estimates of solutions for the initial-boundary value problems are given, and then by using the above estimates of solutions and the Leray-Schauder theorem, the existence and uniqueness of solutions for the problems are proved.展开更多
In this article,we discuss that an oblique derivative boundary value problem for nonlinear uniformly elliptic complex equation of second order with the boundary conditions in a multiply connected unbounded domain D.Th...In this article,we discuss that an oblique derivative boundary value problem for nonlinear uniformly elliptic complex equation of second order with the boundary conditions in a multiply connected unbounded domain D.The above boundary value problem will be called Problem P.Under certain conditions,by using the priori estimates of solutions and Leray-Schauder fixed point theorem,we can obtain some results of the solvability for the above boundary value problem（0.1） and（0.2）.展开更多
This paper deals the irregular oblique derivative problem for some nonlinear elliptic equations of second order. First a priori estimates of solutions are given, afterwards by using the above estimates of solutions an...This paper deals the irregular oblique derivative problem for some nonlinear elliptic equations of second order. First a priori estimates of solutions are given, afterwards by using the above estimates of solutions and the Schauder fixed-point theorem, the existence of solutions for the above boundary value problems is proved.展开更多
The initial\|irregular oblique derivative boundary value problems for linear and nondivergence parabolic complex equations of second order in multiply connected domains are dealt with, where the coefficients of equati...The initial\|irregular oblique derivative boundary value problems for linear and nondivergence parabolic complex equations of second order in multiply connected domains are dealt with, where the coefficients of equations are measurable. Firstly the uniqueness of solutions for the above problems is introduced, and then some \%a priori\% estimates of solutions for the problems are given. By using the above estimates and the Leray\|Schauder theorem, the existence of solutions of the initial\|boundary value problems can be proved. The results are generalizations of corresponding theorems in literature.展开更多
Firstly, the Riemann boundary value problem for a kind of degenerate elliptic sys- tem of the first order equations in R4 is proposed. Then, with the help of the one-to-one correspondence between the theory of Cliffor...Firstly, the Riemann boundary value problem for a kind of degenerate elliptic sys- tem of the first order equations in R4 is proposed. Then, with the help of the one-to-one correspondence between the theory of Clifford valued generalized regular functions and that of the degenerate elliptic system＇s solution, the boundary value problem as stated above is trans- formed into a boundary value problem related to the generalized regular functions in Clifford analysis. Moreover, the solution of the Riemann boundary value problem for the degenerate elliptic system is explicitly described by using a kind of singular integral operator. Finally, the conditions for the existence of solutions of the oblique derivative problem for another kind of degenerate elliptic system of the first order equations in R4 are derived.展开更多
Oblique derivative boundary value problems for the linear mixed (elliptic hyperbolic) equation of the second order, i.e. the generalized Lavrent′ev Bitsadze equation with weak conditions are discussed. The representa...Oblique derivative boundary value problems for the linear mixed (elliptic hyperbolic) equation of the second order, i.e. the generalized Lavrent′ev Bitsadze equation with weak conditions are discussed. The representation of solutions for the above boundary value problem is given, the uniqueness and existence of solutions of the above problem are proved,and a priori estimates of the solutions of the above problem are obtained.展开更多
In this article, the oblique derivative boundary value problems for the mixed type equation uxx＋sgnyuyy=aux＋buy＋cu＋d in another more general domain are discussed and by using the method of complex analysis, a seri...In this article, the oblique derivative boundary value problems for the mixed type equation uxx＋sgnyuyy=aux＋buy＋cu＋d in another more general domain are discussed and by using the method of complex analysis, a series of solvability results is obtained, which generalizes some recent results.展开更多
The present paper deals with oblique derivative problems for second order nonlinear equa- tions of mixed type with degenerate hyperbolic curve, which include the Tricomi problem as a special case. Firstly the formulat...The present paper deals with oblique derivative problems for second order nonlinear equa- tions of mixed type with degenerate hyperbolic curve, which include the Tricomi problem as a special case. Firstly the formulation of the problems for the equations is given, next the representation and estimates of solutions for the above problems are obtained, finally the existence of solutions for the problems is proved by the successive iteration of solutions of the equations and the fixed-point princi- ple. In this paper, we use the complex analytic method, namely the new partial derivative notations, elliptic complex functions in the elliptic domain and hyperbolic complex functions in the hyperbolic domain are introduced, such that the second order equations of mixed type with degenerate curve are reduced to the first order mixed complex equations with singular coefficients, and then the advantage of complex analytic method can be applied.展开更多
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 $$K_1 (y)u_{xx} + \left| {K_2 (x)} \right...The present paper deals with the oblique derivative problem for general second order equations of mixed (elliptic-hyperbolic) type with the nonsmooth parabolic degenerate line $$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 1(y)u x ? iH 2(x)u y]/2 in the elliptic domain and W(z) = W(x+jy) = $u_{\tilde z} $ =[H 1(y)u x ? jH 2(x)u y]/2 in the hyperbolic domain are introduced for the first time, such that the second order equations of mixed type can be reduced to the mixed complex equations of first order with singular coefficients. And thirdly, the advantage of complex analytic method is used, otherwise the complex analytic method cannot be applied.展开更多
This paper deals with the irregular oblique derivative boundary value problemfor nonlinear elliptic systems of second order equations in multiply connected domains.Firstly the well-posedness of the irregUlar oblique d...This paper deals with the irregular oblique derivative boundary value problemfor nonlinear elliptic systems of second order equations in multiply connected domains.Firstly the well-posedness of the irregUlar oblique derivative problem and an a priori es- timate of solutions for the modified boundary value problem are given, afterwards theexistence of solutions for the modified boundary value problem is proved by using theabove estimates of solutions and the Schauder fixed-point theorem. Finally the solVabilityconditions of the original boundary value problem can be derived.展开更多
The authors consider the existence and regularity of the oblique derivative problem:where P is a second order elliptic differential operator on Rn,Ωis a bounded domain in Rn and is a unit vector field on the boundary...The authors consider the existence and regularity of the oblique derivative problem:where P is a second order elliptic differential operator on Rn,Ωis a bounded domain in Rn and is a unit vector field on the boundary of Ω(which may be tangential to the boundary).All above are assumed with limited smoothness. The authors show that solution u has an elliptic gain from f in Holder spaces(Theorem 1.1). The authors obtain LP regualrity of solution in Theorem 1.3, which generalizes the results in [7] to the limited smooth case. Some of the application nonlinear problems are also discussed.展开更多
In this article, we first give the representation of solutions for the oblique derivative problem of mixed （Lavrentév-Bitsadze） equations in two connected domains, afterwards prove the uniqueness of solutions o...In this article, we first give the representation of solutions for the oblique derivative problem of mixed （Lavrentév-Bitsadze） equations in two connected domains, afterwards prove the uniqueness of solutions of the above problem. Moreover, we prove the solvability of oblique derivative problem for quasilinear mixed （Lavrentév-Bitsadze） equations of second order, and obtain a priori estimates of solutions of the above problem. The above problem is an open problem proposed by Rassias.展开更多
This is a continuation of the previous paper [6]. The authors prove Holder and Lp regulaxity of operators collstructed from the oblique derivaive problem in [6] by establishing estimates of pseudodifferential operator...This is a continuation of the previous paper [6]. The authors prove Holder and Lp regulaxity of operators collstructed from the oblique derivaive problem in [6] by establishing estimates of pseudodifferential operators with parameters.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant No. 19671006).
文摘The initial-irregular oblique derivative boundary value problems for nonlinear and nondivergence parabolic systems of second order equations in multiply connected domains are dealt with where coefficients of systems of equations are meaurable. The uniqueness theorem of solutions for the above problems and some a priori estimates of solutions for the problems are given. And by using the above estimates of solutions and the Leray-Schauder theorem, the existence of solutions of the initial-boundary value problems is proved. The results are generalizations of corresponding theorems in literature.
文摘The present article deals with oblique derivative problems for some nonlinear mixed equations with parabolic degeneracy, which include the Tricomi problem as a special case. First, the formulation of the problems for the equations is given; next, the representation and estimates of solutions for the above problems are obtained; finally, the existence of solutions for the problems is proved by the successive iteration and the compactness principle of solutions of the problems. In this article, the author uses the complex method, namely, the complex functions in the elliptic domain and the hyperbolic complex functions in hyperbolic domain are used.
文摘This paper deals with some initial-oblique derivative boundary value problems for nonlinear nondivergent parabolic systems of several second order equations with measurable coefficients in multiply connected domains. Firstly, a priori estimates of solutions for the initial-boundary value problems are given, and then by using the above estimates of solutions and the Leray-Schauder theorem, the existence and uniqueness of solutions for the problems are proved.
文摘In this article,we discuss that an oblique derivative boundary value problem for nonlinear uniformly elliptic complex equation of second order with the boundary conditions in a multiply connected unbounded domain D.The above boundary value problem will be called Problem P.Under certain conditions,by using the priori estimates of solutions and Leray-Schauder fixed point theorem,we can obtain some results of the solvability for the above boundary value problem（0.1） and（0.2）.
文摘This paper deals the irregular oblique derivative problem for some nonlinear elliptic equations of second order. First a priori estimates of solutions are given, afterwards by using the above estimates of solutions and the Schauder fixed-point theorem, the existence of solutions for the above boundary value problems is proved.
文摘The initial\|irregular oblique derivative boundary value problems for linear and nondivergence parabolic complex equations of second order in multiply connected domains are dealt with, where the coefficients of equations are measurable. Firstly the uniqueness of solutions for the above problems is introduced, and then some \%a priori\% estimates of solutions for the problems are given. By using the above estimates and the Leray\|Schauder theorem, the existence of solutions of the initial\|boundary value problems can be proved. The results are generalizations of corresponding theorems in literature.
基金Supported by the National Science Foundation of China(11401162,11571089,11401159,11301136)the Natural Science Foundation of Hebei Province(A2015205012,A2016205218,A2014205069,A2014208158)Hebei Normal University Dr.Fund(L2015B03)
文摘Firstly, the Riemann boundary value problem for a kind of degenerate elliptic sys- tem of the first order equations in R4 is proposed. Then, with the help of the one-to-one correspondence between the theory of Clifford valued generalized regular functions and that of the degenerate elliptic system＇s solution, the boundary value problem as stated above is trans- formed into a boundary value problem related to the generalized regular functions in Clifford analysis. Moreover, the solution of the Riemann boundary value problem for the degenerate elliptic system is explicitly described by using a kind of singular integral operator. Finally, the conditions for the existence of solutions of the oblique derivative problem for another kind of degenerate elliptic system of the first order equations in R4 are derived.
文摘Oblique derivative boundary value problems for the linear mixed (elliptic hyperbolic) equation of the second order, i.e. the generalized Lavrent′ev Bitsadze equation with weak conditions are discussed. The representation of solutions for the above boundary value problem is given, the uniqueness and existence of solutions of the above problem are proved,and a priori estimates of the solutions of the above problem are obtained.
基金Project supported by the EDSF of Shandong Province (J04A11).
文摘In this article, the oblique derivative boundary value problems for the mixed type equation uxx＋sgnyuyy=aux＋buy＋cu＋d in another more general domain are discussed and by using the method of complex analysis, a series of solvability results is obtained, which generalizes some recent results.
基金Supported by National Natural Science Foundation of China （Grant No. 10971224）
文摘The present paper deals with oblique derivative problems for second order nonlinear equa- tions of mixed type with degenerate hyperbolic curve, which include the Tricomi problem as a special case. Firstly the formulation of the problems for the equations is given, next the representation and estimates of solutions for the above problems are obtained, finally the existence of solutions for the problems is proved by the successive iteration of solutions of the equations and the fixed-point princi- ple. In this paper, we use the complex analytic method, namely the new partial derivative notations, elliptic complex functions in the elliptic domain and hyperbolic complex functions in the hyperbolic domain are introduced, such that the second order equations of mixed type with degenerate curve are reduced to the first order mixed complex equations with singular coefficients, and then the advantage of complex analytic method can be applied.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 10671207)
文摘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 1(y)u x ? iH 2(x)u y]/2 in the elliptic domain and W(z) = W(x+jy) = $u_{\tilde z} $ =[H 1(y)u x ? jH 2(x)u y]/2 in the hyperbolic domain are introduced for the first time, such that the second order equations of mixed type can be reduced to the mixed complex equations of first order with singular coefficients. And thirdly, the advantage of complex analytic method is used, otherwise the complex analytic method cannot be applied.
文摘This paper deals with the irregular oblique derivative boundary value problemfor nonlinear elliptic systems of second order equations in multiply connected domains.Firstly the well-posedness of the irregUlar oblique derivative problem and an a priori es- timate of solutions for the modified boundary value problem are given, afterwards theexistence of solutions for the modified boundary value problem is proved by using theabove estimates of solutions and the Schauder fixed-point theorem. Finally the solVabilityconditions of the original boundary value problem can be derived.
文摘The authors consider the existence and regularity of the oblique derivative problem:where P is a second order elliptic differential operator on Rn,Ωis a bounded domain in Rn and is a unit vector field on the boundary of Ω(which may be tangential to the boundary).All above are assumed with limited smoothness. The authors show that solution u has an elliptic gain from f in Holder spaces(Theorem 1.1). The authors obtain LP regualrity of solution in Theorem 1.3, which generalizes the results in [7] to the limited smooth case. Some of the application nonlinear problems are also discussed.
文摘In this article, we first give the representation of solutions for the oblique derivative problem of mixed （Lavrentév-Bitsadze） equations in two connected domains, afterwards prove the uniqueness of solutions of the above problem. Moreover, we prove the solvability of oblique derivative problem for quasilinear mixed （Lavrentév-Bitsadze） equations of second order, and obtain a priori estimates of solutions of the above problem. The above problem is an open problem proposed by Rassias.
文摘This is a continuation of the previous paper [6]. The authors prove Holder and Lp regulaxity of operators collstructed from the oblique derivaive problem in [6] by establishing estimates of pseudodifferential operators with parameters.
