In this paper,the study of gradient regularity for solutions of a class of elliptic problems of p-Laplace type is offered.In particular,we prove a global result concerning Lorentz-Morrey regularity of the non-homogene...In this paper,the study of gradient regularity for solutions of a class of elliptic problems of p-Laplace type is offered.In particular,we prove a global result concerning Lorentz-Morrey regularity of the non-homogeneous boundary data problem:-div((s^(2)+|▽u|^(2)p-2/2)▽u)=-div(|f|^(p-2)f)+g inΩ,u=h in■Ω,with the(sub-elliptic)degeneracy condition s∈[0,1]and with mixed data f∈L^(p)(Q;R^(n)),g∈Lp/(p-1)(Ω;R^(n))for p∈(1,n).This problem naturally arises in various applications such as dynamics of non-Newtonian fluid theory,electro-rheology,radiation of heat,plastic moulding and many others.Building on the idea of level-set inequality on fractional maximal distribution functions,it enables us to carry out a global regularity result of the solution via fractional maximal operators.Due to the significance of M_(α)and its relation with Riesz potential,estimates via fractional maximal functions allow us to bound oscillations not only for solution but also its fractional derivatives of orderα.Our approach therefore has its own interest.展开更多
本文研究了一类具有非局部项的p-Laplace方程的边界最优控制问题,通过对成本泛函的极小化序列取极限给出p-Laplace方程初边值问题最优控制函数的存在性。首先利用能量估计方法研究该问题解的存在唯一性,其次利用紧性估计和紧嵌入定理分...本文研究了一类具有非局部项的p-Laplace方程的边界最优控制问题,通过对成本泛函的极小化序列取极限给出p-Laplace方程初边值问题最优控制函数的存在性。首先利用能量估计方法研究该问题解的存在唯一性,其次利用紧性估计和紧嵌入定理分析成本泛函极小化序列的收敛性,最后由成本泛函的弱下半连续性证明最优控制函数的存在性。In this paper, we study the boundary optimal control problem of a class of p-Laplace equations with non-local terms, and the existence of the optimal control function of the initial boundary value problem of the p-Laplace equation is given by taking the limit of the minimization sequence of the cost function. Firstly, the energy estimation method is used to study the existence uniqueness of the solution of the problem, then the tightness estimation and the tight embedding theorem are used to analyze the convergence of the cost functional minimization sequence, and finally the existence of the optimal control function is proved by the weak lower semi-continuity of the cost function.展开更多
基金supported by Ministry of Education and Training(Vietnam),under grant number B2023-SPS-01。
文摘In this paper,the study of gradient regularity for solutions of a class of elliptic problems of p-Laplace type is offered.In particular,we prove a global result concerning Lorentz-Morrey regularity of the non-homogeneous boundary data problem:-div((s^(2)+|▽u|^(2)p-2/2)▽u)=-div(|f|^(p-2)f)+g inΩ,u=h in■Ω,with the(sub-elliptic)degeneracy condition s∈[0,1]and with mixed data f∈L^(p)(Q;R^(n)),g∈Lp/(p-1)(Ω;R^(n))for p∈(1,n).This problem naturally arises in various applications such as dynamics of non-Newtonian fluid theory,electro-rheology,radiation of heat,plastic moulding and many others.Building on the idea of level-set inequality on fractional maximal distribution functions,it enables us to carry out a global regularity result of the solution via fractional maximal operators.Due to the significance of M_(α)and its relation with Riesz potential,estimates via fractional maximal functions allow us to bound oscillations not only for solution but also its fractional derivatives of orderα.Our approach therefore has its own interest.
文摘本文研究了一类具有非局部项的p-Laplace方程的边界最优控制问题,通过对成本泛函的极小化序列取极限给出p-Laplace方程初边值问题最优控制函数的存在性。首先利用能量估计方法研究该问题解的存在唯一性,其次利用紧性估计和紧嵌入定理分析成本泛函极小化序列的收敛性,最后由成本泛函的弱下半连续性证明最优控制函数的存在性。In this paper, we study the boundary optimal control problem of a class of p-Laplace equations with non-local terms, and the existence of the optimal control function of the initial boundary value problem of the p-Laplace equation is given by taking the limit of the minimization sequence of the cost function. Firstly, the energy estimation method is used to study the existence uniqueness of the solution of the problem, then the tightness estimation and the tight embedding theorem are used to analyze the convergence of the cost functional minimization sequence, and finally the existence of the optimal control function is proved by the weak lower semi-continuity of the cost function.
