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Extremal solution and Liouville theorem for anisotropic elliptic equations 各向异性椭圆方程的极值解和Liouville定理
arXiv: Analysis of PDEs Pub Date : 2021-01-04 DOI: 10.3934/cpaa.2021144
Yuan Li
{"title":"Extremal solution and Liouville theorem for anisotropic elliptic equations","authors":"Yuan Li","doi":"10.3934/cpaa.2021144","DOIUrl":"https://doi.org/10.3934/cpaa.2021144","url":null,"abstract":"We study the quasilinear Dirichlet boundary problem \u0000begin{equation}nonumber left{ begin{aligned} -Qu&=lambda e^{u} quad mbox{in}quadOmega u&=0 quad mbox{on}quadpartialOmega, end{aligned} right. end{equation} where $lambda>0$ is a parameter, $Omegasubsetmathbb{R}^{N}$ with $Ngeq2$ be a bounded domain, and the operator $Q$, known as Finsler-Laplacian or anisotropic Laplacian, is defined by $$Qu:=sum_{i=1}^{N}frac{partial}{partial x_{i}}(F(nabla u)F_{xi_{i}}(nabla u)). $$ Here, $F_{xi_{i}}=frac{partial F}{partialxi_{i}}$ and $F: mathbb{R}^{N}rightarrow[0,+infty)$ is a convex function of $ C^{2}(mathbb{R}^{N}setminus{0})$, that satisfies certain assumptions. We derive the existence of extremal solution and obtain that it's regular, if $Nleq9$. \u0000We also concern the H'{e}non type anisotropic Liouville equation, namely, $$-Qu=(F^{0}(x))^{alpha}e^{u}quadmbox{in}quadmathbb{R}^{N}$$ where $alpha>-2$, $Ngeq2$ and $F^{0}$ is the support function of $K:={xinmathbb{R}^{N}:F(x)<1}$ which is defined by $$F^{0}(x):=sup_{xiin K}langle x,xirangle.$$ We obtain the Liouville theorem for stable solutions and the finite Morse index solutions for $2leq N<10+4alpha$ and $3leq N<10+4alpha^{-}$ respectively, where $alpha^{-}=min{alpha,0}$.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79799253","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Entropy inequality and energy dissipation of inertial Qian–Sheng model for nematic liquid crystals 向列液晶惯性钱生模型的熵不等式和能量耗散
arXiv: Analysis of PDEs Pub Date : 2020-12-14 DOI: 10.1142/S0219891621500065
Ning Jiang, Yi-Long Luo, Yangjun Ma, Shaojun Tang
{"title":"Entropy inequality and energy dissipation of inertial Qian–Sheng model for nematic liquid crystals","authors":"Ning Jiang, Yi-Long Luo, Yangjun Ma, Shaojun Tang","doi":"10.1142/S0219891621500065","DOIUrl":"https://doi.org/10.1142/S0219891621500065","url":null,"abstract":"For the inertial Qian-Sheng model of nematic liquid crystals in the $Q$-tensor framework, we illustrate the roles played by the entropy inequality and energy dissipation in the well-posedness of smooth solutions when we employ energy method. We first derive the coefficients requirements from the entropy inequality, and point out the entropy inequality is insufficient to guarantee energy dissipation. We then introduce a novel Condition (H) which ensures the energy dissipation. We prove that when both the entropy inequality and Condition (H) are obeyed, the local in time smooth solutions exist for large initial data. Otherwise, we can only obtain small data local solutions. Furthermore, to extend the solutions globally in time and obtain the decay of solutions, we require at least one of the two conditions: entropy inequality, or $tilde{mu}_2= mu_2$, which significantly enlarge the range of the coefficients in previous works.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84958690","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Existence and uniqueness of axially symmetric compressible subsonic jet impinging on an infinite wall 轴对称可压缩亚音速射流撞击无限大壁面的存在唯一性
arXiv: Analysis of PDEs Pub Date : 2020-12-13 DOI: 10.4171/IFB/449
Jianfeng Cheng, Lili Du, Qin Zhang
{"title":"Existence and uniqueness of axially symmetric compressible subsonic jet impinging on an infinite wall","authors":"Jianfeng Cheng, Lili Du, Qin Zhang","doi":"10.4171/IFB/449","DOIUrl":"https://doi.org/10.4171/IFB/449","url":null,"abstract":"This paper is concerned with the well-posedness theory of the impact of a subsonic axially symmetric jet emerging from a semi-infinitely long nozzle, onto a rigid wall. The fluid motion is described by the steady isentropic Euler system. We showed that there exists a critical value $M_{cr}>0$, if the given mass flux is less than $M_{cr}$, there exists a unique smooth subsonic axially symmetric jet issuing from the given semi-infinitely long nozzle and hitting a given uneven wall. The surface of the axially symmetric impinging jet is a free boundary, which detaches from the edge of the nozzle smoothly. It is showed that a unique suitable choice of the pressure difference between the chamber and the atmosphere guarantees the continuous fit condition of the free boundary. Moreover, the asymptotic behaviors and the decay properties of the impinging jet and the free surface in downstream were also obtained. The main results in this paper solved the open problem on the well-posedness of the compressible axially symmetric impinging jet, which has proposed by A. Friedman in Chapter 16 in [FA2]. The key ingredient of our proof is based on the variational method to the quasilinear elliptic equation with the Bernoulli's type free boundaries.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74483939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Sharp critical thresholds in a hyperbolic system with relaxation 带松弛的双曲系统的尖锐临界阈值
arXiv: Analysis of PDEs Pub Date : 2020-12-12 DOI: 10.3934/dcds.2021098
Manas Bhatnagar, Hailiang Liu
{"title":"Sharp critical thresholds in a hyperbolic system with relaxation","authors":"Manas Bhatnagar, Hailiang Liu","doi":"10.3934/dcds.2021098","DOIUrl":"https://doi.org/10.3934/dcds.2021098","url":null,"abstract":"We propose and study a one-dimensional $2times 2$ hyperbolic Eulerian system with local relaxation from critical threshold phenomena perspective. The system features dynamic transition between strictly and weakly hyperbolic. For different classes of relaxation we identify intrinsic critical thresholds for initial data that distinguish global regularity and finite time blowup. For relaxation independent of density, we estimate bounds on density in terms of velocity where the system is strictly hyperbolic.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82822479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
On instability mechanisms for inverse problems 反问题的不稳定性机制
arXiv: Analysis of PDEs Pub Date : 2020-12-03 DOI: 10.15781/c93s-pk62
H. Koch, Angkana Ruland, M. Salo
{"title":"On instability mechanisms for inverse problems","authors":"H. Koch, Angkana Ruland, M. Salo","doi":"10.15781/c93s-pk62","DOIUrl":"https://doi.org/10.15781/c93s-pk62","url":null,"abstract":"In this article we present three robust instability mechanisms for linear and nonlinear inverse problems. All of these are based on strong compression properties (in the sense of singular value or entropy number bounds) which we deduce through either strong global smoothing, only weak global smoothing or microlocal smoothing for the corresponding forward operators, respectively. As applications we for instance present new instability arguments for unique continuation, for the backward heat equation and for linear and nonlinear Calderon type problems in general geometries, possibly in the presence of rough coefficients. Our instability mechanisms could also be of interest in the context of control theory, providing estimates on the cost of (approximate) controllability in rather general settings.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83701032","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 17
Existence of positive solutions for a singular elliptic problem with critical exponent and measure data 一类具有临界指数和测量数据的奇异椭圆型问题正解的存在性
arXiv: Analysis of PDEs Pub Date : 2020-12-01 DOI: 10.1216/rmj.2021.51.973
A. Panda, D. Choudhuri, R. K. Giri
{"title":"Existence of positive solutions for a singular elliptic problem with critical exponent and measure data","authors":"A. Panda, D. Choudhuri, R. K. Giri","doi":"10.1216/rmj.2021.51.973","DOIUrl":"https://doi.org/10.1216/rmj.2021.51.973","url":null,"abstract":"We prove the existence of a positive {it SOLA (Solutions Obtained as Limits of Approximations)} to the following PDE involving fractional power of Laplacian begin{equation} begin{split} (-Delta)^su&= frac{1}{u^gamma}+lambda u^{2_s^*-1}+mu ~text{in}~Omega, u&>0~text{in}~Omega, u&= 0~text{in}~mathbb{R}^NsetminusOmega. end{split} end{equation} Here, $Omega$ is a bounded domain of $mathbb{R}^N$, $sin (0,1)$, $2s<N$, $lambda,gammain (0,1)$, $2_s^*=frac{2N}{N-2s}$ is the fractional critical Sobolev exponent and $mu$ is a nonnegative bounded Radon measure in $Omega$.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76308152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Nonlocal characterizations of variable exponent Sobolev spaces 变指数Sobolev空间的非局部刻画
arXiv: Analysis of PDEs Pub Date : 2020-12-01 DOI: 10.3233/ASY-211675
G. Ferrari, M. Squassina
{"title":"Nonlocal characterizations of variable exponent Sobolev spaces","authors":"G. Ferrari, M. Squassina","doi":"10.3233/ASY-211675","DOIUrl":"https://doi.org/10.3233/ASY-211675","url":null,"abstract":"We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity theory and in the theory of electrorheological fluids. We also get a singular limit formula extending Nguyen results to the anisotropic case.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81003414","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Time quasi-periodic traveling gravity water waves in infinite depth 时间准周期行重力水波在无限深度
arXiv: Analysis of PDEs Pub Date : 2020-11-24 DOI: 10.4171/RLM/919
R. Feola, Filippo Giuliani
{"title":"Time quasi-periodic traveling gravity water waves in infinite depth","authors":"R. Feola, Filippo Giuliani","doi":"10.4171/RLM/919","DOIUrl":"https://doi.org/10.4171/RLM/919","url":null,"abstract":"We present the recent result [8] concerning the existence of quasi-periodic in time traveling waves for the 2d pure gravity water waves system in infinite depth. We provide the first existence result of quasi-periodic water waves solutions bifurcating from a completely resonant elliptic fixed point. The proof is based on a Nash-Moser scheme, Birkhoff normal form methods and pseudo-differential calculus techniques. We deal with the combined problems of small divisors and the fully-nonlinear nature of the equations.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74366364","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
A class of functionals possessing multiple global minima 一类具有多个全局极小值的泛函
arXiv: Analysis of PDEs Pub Date : 2020-11-24 DOI: 10.24193/SUBBMATH.2021.1.06
B. Ricceri
{"title":"A class of functionals possessing multiple global minima","authors":"B. Ricceri","doi":"10.24193/SUBBMATH.2021.1.06","DOIUrl":"https://doi.org/10.24193/SUBBMATH.2021.1.06","url":null,"abstract":"We get a new multiplicity result for gradient systems. Here is a very particular corollary: Let $Omegasubset {bf R}^n$ ($ngeq 2$) be a smooth bounded domain and let $Phi:{bf R}^2to {bf R}$ be a $C^1$ function, with $Phi(0,0)=0$, such that $$sup_{(u,v)in {bf R}^2}{{|Phi_u(u,v)|+|Phi_v(u,v)|}over {1+|u|^p+|v|^p}} 0$, with $p 2$. Then, for every convex set $Ssubseteq L^{infty}(Omega)times L^{infty}(Omega)$ dense in $L^2(Omega)times L^2(Omega)$, there exists $(alpha,beta)in S$ such that the problem \u0000$$cases {-Delta u=(alpha(x)cos(Phi(u,v))-beta(x)sin(Phi(u,v)))Phi_u(u,v) & in $Omega$ cr & cr -Delta v= (alpha(x)cos(Phi(u,v))-beta(x)sin(Phi(u,v)))Phi_v(u,v) & in $Omega$ cr & cr u=v=0 & on $partialOmega$cr}$$ has at least three weak solutions, two of which are global minima in $H^1_0(Omega)times H^1_0(Omega)$ of the functional $$(u,v)to {{1}over {2}}left ( int_{Omega}|nabla u(x)|^2dx+int_{Omega}|nabla v(x)|^2dxright )$$ $$-int_{Omega}(alpha(x)sin(Phi(u(x),v(x)))+beta(x)cos(Phi(u(x),v(x))))dx .$$","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74946142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Singular solutions to parabolic equations in nondivergence form 非发散形式抛物方程的奇异解
arXiv: Analysis of PDEs Pub Date : 2020-11-23 DOI: 10.2422/2036-2145.202011_110
L. Silvestre
{"title":"Singular solutions to parabolic equations in nondivergence form","authors":"L. Silvestre","doi":"10.2422/2036-2145.202011_110","DOIUrl":"https://doi.org/10.2422/2036-2145.202011_110","url":null,"abstract":"For any $alpha in (0,1)$, we construct an example of a solution to a parabolic equation with measurable coefficients in two space dimensions which has an isolated singularity and is not better that $C^alpha$. We prove that there exists no solution to a fully nonlinear uniformly parabolic equation, in any dimension, which has an isolated singularity where it is not $C^2$ while it is analytic elsewhere, and it is homogeneous in $x$ at the time of the singularity. We build an example of a non homogeneous solution to a fully nonlinear uniformly parabolic equation with an isolated singularity, which we verify with the aid of a numerical computation.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80245568","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
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