# 一类与分数阶p-Laplace算子相关的重排优化问题An Optimization Problem Involving the p-Fractional Laplacian

Abstract: This paper focuses on an optimization problem involving the fractional p-Laplacian. Firstly, we use the global minimum principle in the suitable variational framework to obtain a global minimum solution of a fractional p-Laplacian equation. Then, the uniqueness of the solution of the equation can be obtained by using reduction to absurdity. Finally, the solvability of a minimization problem for the energy functional corresponding to the fractional p-Laplace equation will be verified by rearrangement optimization theory.

1. 引言

$\left\{\begin{array}{l}{\left(-\Delta \right)}_{p}^{s}u-\lambda V\left(x\right){|u|}^{p-2}u=f\left(x\right),\text{}x\in \Omega \\ u=0,x\in {R}^{N}\\Omega \end{array}$

${\left(-\Delta \right)}_{p}^{s}u\left(x\right)=2\underset{\epsilon ↓0}{\mathrm{lim}}{\int }_{{R}^{N}\{B}_{\epsilon }\left(x\right)}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)}{{|x-y|}^{N+ps}}\text{d}y,$

(Opt) $\underset{g\in R\left(f\right)}{\mathrm{inf}}\Phi \left( g \right)$

$\Phi \left(g\right)=\frac{1}{p}{‖{u}_{g}‖}^{p}-\frac{\lambda }{p}{\int }_{\Omega }V\left(x\right){|{u}_{g}|}^{p}\text{d}x-{\int }_{\Omega }g{u}_{g}\text{d}x$

2. 预备知识

$\Omega \subset {R}^{N},\text{}\left(N\ge 3\right)$ 为一个有界光滑区域。设函数 $f:\Omega \to R$ 可测，我们记 $R\left(f\right)$ 为由满足如下条件：

$meas\left(\left\{x\in \Omega :g\left(x\right)\ge a\right\}\right)=meas\left(\left\{x\in \Omega :f\left(x\right)\ge a\right\}\right),\text{}\forall a\in R$

${\left[u\right]}_{s,p}={\left({\int }_{{R}^{N}}{\int }_{{R}^{N}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p}}{{|x-y|}^{N+ps}}\text{d}x\text{d}y\right)}^{1/p}$

${W}_{}^{s,p}\left({R}^{N}\right)=\left\{u\in {L}^{p}\left({R}^{N}\right):{\left[u\right]}_{s,p}<\infty \right\}$

${‖u‖}_{s,p}={\left({\int }_{{R}^{N}}{|u|}^{p}\text{d}x+{\left[u\right]}_{s,p}^{p}\right)}^{1/p}$

${W}_{0}^{s,p}\left(\Omega \right)=\left\{u\in {W}^{s,p}\left({R}^{N}\right):u\equiv 0,x\in {R}^{N}\\Omega \right\},$

${I}_{f}\left(u\right)=\frac{1}{p}{\int }_{{R}^{N}}{\int }_{{R}^{N}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p}}{{|x-y|}^{N+ps}}\text{d}x\text{d}y-\frac{\lambda }{p}{\int }_{\Omega }V{|u|}^{p}\text{d}x-{\int }_{\Omega }fu\text{d}x$

$〈{{I}^{\prime }}_{f}\left(u\right),v〉={\int }_{{R}^{N}}{\int }_{{R}^{N}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left(v\left(x\right)-v\left(y\right)\right)}{{|x-y|}^{N+ps}}\text{d}x\text{d}y-\lambda {\int }_{\Omega }V{|u|}^{p-2}uv\text{d}x-{\int }_{\Omega }fv\text{d}x$

${\int }_{{R}^{N}}{\int }_{{R}^{N}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)\left(v\left(x\right)-v\left(y\right)\right)}{{|x-y|}^{N+ps}}\text{d}x\text{d}y-\lambda {\int }_{\Omega }V{|u|}^{p-2}uv\text{d}x-{\int }_{\Omega }fv\text{d}x=0$

${\lambda }_{1}=\underset{u\in {W}_{0}^{s,p}\left(\Omega \right)}{\mathrm{inf}}\left\{{‖u‖}^{p}:{\int }_{\Omega }V{|u|}^{p}\text{d}x=1\right\}$ (2.1)

${\int }_{\Omega }V{|{u}_{n}|}^{p}\text{d}x=1,\text{}{‖{u}_{n}‖}^{p}\to {\lambda }_{1},$

${\int }_{\Omega }V|\left({|{u}_{n}|}^{p}-{|\stackrel{¯}{u}|}^{p}\right)|\text{d}x\le {‖V‖}_{{L}^{\infty }}{\int }_{\Omega }|\left({|{u}_{n}|}^{p}-{|\stackrel{¯}{u}|}^{p}\right)|\text{d}x\to 0$

${\int }_{\Omega }V{|{u}_{n}|}^{p}\text{d}x\to {\int }_{\Omega }V{|\stackrel{¯}{u}|}^{p}\text{d}x=1$

${\lambda }_{1}\le {‖\stackrel{¯}{u}‖}^{p}\le \underset{n\to \infty }{\mathrm{lim}\mathrm{inf}}{‖{u}_{n}‖}^{p}={\lambda }_{1}$

3. 方程 $\left({P}_{\lambda ,f}\right)$ 全局极小解的存在唯一性

$〈{{I}^{\prime }}_{f}\left({u}_{f}\right),v〉=0,\text{}\forall v\in {W}_{0}^{s,p}\left( \Omega \right)$

$I\left({u}_{f}\right)={\mathrm{inf}}_{v\in {W}_{0}^{s,p}\left(\Omega \right)}I\left( v \right)$

$|{\int }_{\Omega }fu\text{d}x|\le {‖f‖}_{{L}^{q}}{‖u‖}_{{L}_{q\text{'}}}\le C‖u‖$

${I}_{f}\left(u\right)\ge \frac{1}{p}\left(1-\frac{\lambda }{{\lambda }_{1}}\right){‖u‖}^{p}-C‖u‖\to \infty ,\text{}若‖u‖\to \infty$

$〈{{I}^{\prime }}_{f}\left({u}_{f}\right),v〉=0,\forall v\in {W}_{0}^{s,p}\left( \Omega \right)$

$m:={\mathrm{inf}}_{v\in {W}_{0}^{s,p}\left(\Omega \right)}{I}_{f}\left(v\right)$，则 ${I}_{f}\left({u}_{f}\right)=m$。由于 $0<\lambda <{\lambda }_{1}$，则

${‖u‖}^{p}\ge {‖u‖}^{p}-\lambda {\int }_{\Omega }V{|u|}^{p}\text{d}x\ge \frac{{\lambda }_{1}-\lambda }{{\lambda }_{1}}{‖u‖}^{p}$

$t\in \left(0,1\right),\text{}u,v\in {W}_{0}^{s,p}\left(\Omega \right)$ 成立

${‖tu+\left(1-t\right)v‖}_{\lambda }\le t{‖u‖}_{\lambda }+\left(1-t\right){‖v‖}_{\lambda }$

$\begin{array}{l}{I}_{f}\left(tu+\left(1-t\right)v\right)\\ =\frac{1}{p}{‖tu+\left(1-t\right)v‖}_{\lambda }^{p}-{\int }_{\Omega }f\left(x\right)\left(tu+\left(1-t\right)v\right)\text{d}x\\ \le \frac{1}{p}{\left(t{‖u‖}_{\lambda }^{}+\left(1-t\right){‖v‖}_{\lambda }^{}\right)}^{p}-{\int }_{\Omega }f\left(x\right)\left(tu+\left(1-t\right)v\right)\text{d}x\\ <\frac{t}{p}{‖u‖}_{\lambda }^{p}+\frac{1-t}{p}{‖v‖}_{\lambda }^{p}-{\int }_{\Omega }f\left(x\right)\left(tu+\left(1-t\right)v\right)\text{d}x\\ =t{I}_{f}\left(u\right)+\left(1-t\right){I}_{f}\left( v \right)\end{array}$

4. 能量泛函重排优化问题(Opt)极小点的存在性

$\Phi \left(\stackrel{^}{f}\right)={I}_{\stackrel{^}{f}}\left(\stackrel{^}{u}\right)=\underset{g\in R\left(f\right)}{\mathrm{inf}}{I}_{g}\left({u}_{g}\right)=\underset{g\in R\left(f\right)}{\mathrm{inf}}\Phi \left( g \right)$

$\begin{array}{c}{I}_{g}\left({u}_{g}\right)=\frac{1}{p}{\int }_{{R}^{N}}{\int }_{{R}^{N}}\frac{{|{u}_{g}\left(x\right)-{u}_{g}\left(y\right)|}^{p}}{{|x-y|}^{N+ps}}\text{d}x\text{d}y-\frac{\lambda }{p}{\int }_{\Omega }V{|{u}_{g}|}^{p}\text{d}x-{\int }_{\Omega }g{u}_{g}\text{d}x\\ \ge \frac{1}{p}\left(1-\frac{\lambda }{{\lambda }_{1}}\right){‖{u}_{g}‖}^{p}-C{‖g‖}_{{L}_{q}}‖{u}_{g}‖。\end{array}$ (4.1)

${g}_{i}\in R\left(f\right)$$\forall i\in N$

$A=\underset{i\to \infty }{\mathrm{lim}}I\left( u i \right)$

${L}^{{q}^{\prime }}\left(\Omega \right)\left(1<{q}^{\prime }=q/\left(q-1\right) 可知其在 ${L}^{{q}^{\prime }}\left(\Omega \right)$ 中强收敛于 $u$。令 ${\stackrel{¯}{R\left(f\right)}}^{q,w}$$R\left(f\right)$${L}^{q}\left(\Omega \right)$ 中的

$|{\int }_{\Omega }\left({g}_{i}-\stackrel{¯}{g}\right)u\text{d}x|\to 0,\text{}i\to \infty$

$\begin{array}{c}|{\int }_{\Omega }\left({g}_{i}{u}_{i}-\stackrel{¯}{g}u\right)\text{d}x|\le |{\int }_{\Omega }{g}_{i}\left({u}_{i}-u\right)\text{d}x|+|{\int }_{\Omega }\left({g}_{i}-\stackrel{¯}{g}\right)u\text{d}x|\\ \le {‖{g}_{i}‖}_{{L}^{q}}{‖{u}_{i}-u‖}_{{L}^{{q}^{\prime }}}+|{\int }_{\Omega }\left({g}_{i}-\stackrel{¯}{g}\right)u\text{d}x|\to 0,i\to \infty 。\end{array}$

$\underset{i\to \infty }{\mathrm{lim}}{\int }_{\Omega }V\left({|{u}_{i}|}^{p}-{|u|}^{p}\right)\text{d}x=0$

$A=\underset{i\to \infty }{\mathrm{lim}}I\left({u}_{i}\right)\ge \frac{1}{p}{\int }_{{R}^{N}}{\int }_{{R}^{N}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p}}{{|x-y|}^{N+ps}}\text{d}x\text{d}y-\frac{\lambda }{p}{\int }_{\Omega }V{|u|}_{}^{p}\text{d}x-{\int }_{\Omega }\stackrel{¯}{g}u\text{d}x$ (4.2)

${\int }_{\Omega }\stackrel{¯}{g}u\text{d}x\le {\int }_{\Omega }\stackrel{^}{f}u\text{d}x$

$A\ge \frac{1}{p}{\int }_{{R}^{N}}{\int }_{{R}^{N}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p}}{{|x-y|}^{N+ps}}\text{d}x\text{d}y-\frac{\lambda }{p}{\int }_{\Omega }V{|u|}_{}^{p}\text{d}x-{\int }_{\Omega }\stackrel{^}{f}u\text{d}x$ (4.3)

$\begin{array}{c}I\left(\stackrel{^}{u}\right)=\underset{v\in {W}_{0}^{s,p}\left(\Omega \right)}{\mathrm{inf}}\frac{1}{p}{\int }_{{R}^{N}}{\int }_{{R}^{N}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p}}{{|x-y|}^{N+ps}}\text{d}x\text{d}y-\frac{\lambda }{p}{\int }_{\Omega }V{|v|}_{}^{p}\text{d}x-{\int }_{\Omega }\stackrel{^}{f}v\text{d}x\\ \le \frac{1}{p}{\int }_{{R}^{N}}{\int }_{{R}^{N}}\frac{{|u\left(x\right)-u\left(y\right)|}^{p}}{{|x-y|}^{N+ps}}\text{d}x\text{d}y-\frac{\lambda }{p}{\int }_{\Omega }V{|u|}_{}^{p}\text{d}x-{\int }_{\Omega }\stackrel{^}{f}u\text{d}x。\end{array}$ (4.4)

$I\left(\stackrel{^}{u}\right)=A$

[1] Burton, G.R. (1987) Rearrangements of Functions, Maximization of Convex Functionals and Vortex Rings. Mathematische Annalen, 276, 225-253.
https://doi.org/10.1007/BF01450739

[2] Burton, G.R. (1989) Variational Problems on Classes of Rearrangements and Multiple Configurations for Steady Vortices. Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 6, 295-319.
https://doi.org/10.1016/S0294-1449(16)30320-1

[3] Cuccu, F., Emamizadeh, B. and Porru, G. (2006) Nonlinear Elastic Membrane Involving the p-Laplacian Operator. Electronic Journal of Differential Equations, 2006, 1-10.

[4] Cuccu, F., Emamizadeh, B. and Porru, G. (2009) Optimization of the First Eigenvalue in Problems Involving the p- Laplacian. Proceedings of the American Mathematical Society, 137, 1677-1687.
https://doi.org/10.1090/S0002-9939-08-09769-4

[5] Cuccu, F., Porru, G. and Sakaguchi, S. (2011) Optimization Problems on General Classes of Rearrangements. Nonlinear Analysis: Theory, Methods & Applications, 74, 5554-5565.
https://doi.org/10.1016/j.na.2011.05.039

[6] Emamizadeh, B. and Zivari-Rezapour, M. (2007) Rearrangement Optimization for Some Elliptic Equations. Journal of Optimization Theory and Applications, 135, 367-379.
https://doi.org/10.1007/s10957-007-9269-y

[7] Emamizadeh, B. and Prajapat, J.V. (2009) Symmetry in Rearrangemet Optimization Problems. Electronic Journal of Differential Equations, 2009, 1-10.

[8] Marras, M. (2010) Optimization in Problems Involving the p-Laplacian. Electronic Journal of Differential Equations, 2010, 1-10.

[9] Qiu, C., Huang, Y.S. and Zhou, Y.Y. (2015) A Class of Rearrangement Optimization Problems Involving the p-Laplacian. Nonlinear Analysis: Theory, Methods & Applications, 112, 30-42.
https://doi.org/10.1016/j.na.2014.09.008

[10] Qiu, C., Huang, Y.S. and Zhou, Y.Y. (2016) Optimization Problems Involving the Fractional Laplacian. Electronic Journal of Differential Equations, 2016, 1-15.

[11] Dalibard, A.L. and Gerard-Varet, D. (2013) On Shape Optimization Problems Involving the Fractional Laplacian. ESAIM: Control, Optimisation and Calculus of Variations, 19, 976-1013.
https://doi.org/10.1051/cocv/2012041

[12] Biswas, A. and Jarohs, S. (2020) On Over-Determined Problems for a General Class of Nonlocal Operators. Journal of Differential Equations, 268, 2368-2393.
https://doi.org/10.1016/j.jde.2019.09.010

[13] Bonder, J.F., Ritorto, A. and Salort, A.M. (2018) A Class of Shape Optimization Problems for Nonlocal Operators. Advances in Calculus of Variations, 11, 373-386.
https://doi.org/10.1515/acv-2016-0065

[14] Bonder, J.F., Rossi, J.D. and Spedaletti, J.F. (2018) Optimal Design Problems for the First p-Fractional Eigenvalue with Mixed Boundary Conditions. Advanced Nonlinear Studies, 18, 323-335.
https://doi.org/10.1515/ans-2018-0001

[15] Pezzo, L.D., Bonder, J.F. and Rios, L.L. (2018) An Optimization Problem for the First Eigenvalue of the p-Fractional Laplacian. Mathematische Nachrichten, 291, 632-651.
https://doi.org/10.1002/mana.201600110

[16] Bonder, J.F. and Spedaletti, J.F. (2018) Some Nonlocal Optimal Design Problems. Journal of Mathematical Analysis and Applications, 459, 906-931.
https://doi.org/10.1016/j.jmaa.2017.11.015

[17] Di Nezza, E., Palatucci, G. and Valdinoci, E. (2012) Hitchhiker’s Guide to the Fractional Sobolev Spaces. Bulletin of Mathematical Sciences, 136, 521-573.
https://doi.org/10.1016/j.bulsci.2011.12.004

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