Recent zbMATH articles in MSC 31Bhttps://www.zbmath.org/atom/cc/31B2021-11-25T18:46:10.358925ZWerkzeugPluripotential theory on Teichmüller space. I: Pluricomplex Green functionhttps://www.zbmath.org/1472.300182021-11-25T18:46:10.358925Z"Miyachi, Hideki"https://www.zbmath.org/authors/?q=ai:miyachi.hidekiIn the paper [Bull. Am. Math. Soc., New Ser. 27, No. 1, 143--147 (1992; Zbl 0766.30016)], \textit{S. L. Krushkal} announced the following result on the pluricomplex Green function on Teichmüller space:
Theorem. Let \(T_{g, m}\) be the Teichmüller space of Riemann surfaces of analytically finite-type \((g,m)\), and let \(d_T\) be the Teichmüller distance on \(T_{g, m}\). Then, the pluricomplex Green function \(g_{T_{g, m}} \) on \(T_{g, m}\) satisfies \[g_{T_{g, m}}(x, y)\log \tanh d_T(x, y)\] for \(x, y \in g_{T_{g, m}}\).
The author has a programm to investigate of the pluripotential theory on Teichmüller space. In this first one of a series of works, the author gave an alternative approach to the Krushkal formula of the pluricomplex Green function on Teichmüller space. In comparison with the original approach by Krushkal, the strategy is more direct here. He first shows that the Teichmüller space carries a natural stratified structure of real-analytic submanifolds defined from the structure of singularities of the initial differentials of the Teichmüller mappings from a given point. Then he gives a description of the Levi form of the pluricomplex Green function using the Thurston symplectic form via Dumas' symplectic structure on the space of holomorphic quadratic differentials [\textit{D. Dumas}, Acta Math. 215, No. 1, 55--126 (2015; Zbl 1334.57020)].The family of level sets of a harmonic functionhttps://www.zbmath.org/1472.310032021-11-25T18:46:10.358925Z"Ding, Pisheng"https://www.zbmath.org/authors/?q=ai:ding.pishengThe main result, Theorem 1, characterizes families of level-sets of harmonic functions of two variables without critical points in terms of the curvature \(\kappa\) of a level set \(\gamma_t\). The proof relies on the basic fact (Lemma 3) that a \(C^2\) function \(f\) on a domain \(\Omega\subset \mathbb{R}^2\) without critical points is harmonic if and only if \(\kappa\equiv \frac{D_N^2 f}{D_Nf}\) on \(\Omega\), where \(N=\nabla f/|\nabla f|\) is the unit normal field, \(D_N f\), \(D_N^2 f\) are the first and second directional derivatives of \(f\) along \(N\).
The author also extends Theorem 1 to higher dimensions for the case when the level sets are diffeomorphic to \(\mathbb{R}^{n-1}\) (Theorem 4).A proof of the Khavinson conjecturehttps://www.zbmath.org/1472.310072021-11-25T18:46:10.358925Z"Liu, Congwen"https://www.zbmath.org/authors/?q=ai:liu.congwenThe author gives a complete proof of the validity of the Khavinson conjecture. In order to state the conjecture, let \(h^\infty\) be the space of bounded harmonic functions on the unit ball \(\mathbb{B}^n\) of \(\mathbb{R}^n\), with \(n \geq 3\). For \(x \in \mathbb{B}^n\) we denote by \(C(x)\) the smallest number such that
\[
|\nabla u(x)| \leq C(x)\sup_{y \in \mathbb{B}^n}|u(y)|
\]
for all \(u \in h^\infty\). Similarly, for \(x\in \mathbb{B}^n\) and \(l\in \partial \mathbb{B}^n\), we denote by \(C(x,l)\) the smallest number such that
\[
|\langle\nabla u(x),l \rangle | \leq C(x,l)\sup_{y \in \mathbb{B}^n}|u(y)|
\]
for all \(u \in h^\infty\). As it is well known, both constants are finite. The Khavinson conjecture states that for \(x \in \mathbb{B}^n \setminus \{0\}\) we have
\[
C(x)=C\left(x,\frac{x}{|x|}\right)\, .
\]
The author shows the validity of the conjecture, by considering an equivalent optimization problem and by solving such a problem in terms of the Gegenbauer polynomials.Weak estimates for the maximal and Riesz potential operators in central Herz-Morrey spaces on the unit ballhttps://www.zbmath.org/1472.310082021-11-25T18:46:10.358925Z"Mizuta, Yoshihiro"https://www.zbmath.org/authors/?q=ai:mizuta.yoshihiro"Ohno, Takao"https://www.zbmath.org/authors/?q=ai:ohno.takao"Shimomura, Tetsu"https://www.zbmath.org/authors/?q=ai:shimomura.tetsuThis nice paper introduces the weak central Herz-Morrey spaces \(WH^{p(\cdot),q,\omega}(\mathbf B)\) and \(WH^{p^\ast(\cdot),q,\omega}(\mathbf B)\) with \( p^\ast(x) = p(x)N/(N-\alpha p(x))\) on the Euclidean unit ball \(\mathbf B\) and shows the boundedness of the generalized maximal operator \(M_\beta\) and the Riesz potential operator \(I_\alpha\) from the non-homogeneous central Herz-Morrey space \(H^{p(\cdot),q,\omega}(\mathbf B)\) to \(WH^{p(\cdot),q,\omega}(\mathbf B)\) (Theorem 3.10) and \(WH^{p^\ast(\cdot),q,\omega}(\mathbf B)\) (Theorem 4.1), respectively.Approximate tangents, harmonic measure, and domains with rectifiable boundarieshttps://www.zbmath.org/1472.310092021-11-25T18:46:10.358925Z"Mourgoglou, Mihalis"https://www.zbmath.org/authors/?q=ai:mourgoglou.mihalisThis article discusses the connection between approximate tangents, harmonic measures and domains with rectifiable boundaries. In the first result of the paper it is established that if \(E\subset\mathbb{R}^{n+1}\) is closed, \(0<s<1/3\) and \(\mathcal{T}_m(E)\subset E\) be the set of all points \(x\in E\) such that:
(i) there exists an \(s\)-approximate tangent \(m\)-plane \(V_x\) for \(E\) at \(x\);
(ii) \(E\) satisfies the weak lower Ahlfors-David \(m\)-regularity condition at \(x\).
Then, there exists a countable collection of bounded Lipschitz graphs \(\{\Gamma_j\}_{j\geq 1}\) so that \(\mathcal{T}_m(E)\subset \bigcup_{j\geq 1}\Gamma_j\). In particular, \(\mathcal{T}_m(E)\) is \(m\)-rectifiable.
In the second result of the article it is obtained that if \(0<s<1/\sqrt{90}\), then there exist two countable collections of bounded Lipschitz domains \(\{\Omega_j^\pm\}_{j\geq 1}\) such that \(\Omega_j^+\cap \Omega_j^-=\emptyset\), \(\mathcal{T}_n(E)\cap \Omega_j^+=\mathcal{T}_n\cap\Omega_j^-\) and \(\mathcal{T}_m(E)\subset \bigcup_{j\geq 1}\Omega_j^\pm\).
Further characterizations of the countable collections of bounded Lipschitz domains \(\{\Omega_j^\pm\}_{j\geq 1}\) are provided in the article.Duality between range and no-response tests and its application for inverse problemshttps://www.zbmath.org/1472.310102021-11-25T18:46:10.358925Z"Lin, Yi-Hsuan"https://www.zbmath.org/authors/?q=ai:lin.yi-hsuan"Nakamura, Gen"https://www.zbmath.org/authors/?q=ai:nakamura.gen"Potthast, Roland"https://www.zbmath.org/authors/?q=ai:potthast.roland-w-e"Wang, Haibing"https://www.zbmath.org/authors/?q=ai:wang.haibingThe authors show the duality between range and no-response tests for an inverse boundary value problem for the Laplace equation in \(\Omega \setminus \overline{D}\) with an unknown obstacle \(D\) whose closure is contained in \(\Omega\). They consider the boundary value problem
\[
\left\{ \begin{array}{ll} \Delta u=0 & \mbox{in}\ \Omega \setminus \overline{D}\, ,\\
u=0 & \mbox{on}\ \partial D \, ,\\
u=f & \mbox{on}\ \partial \Omega \, . \end{array} \right.
\]
The Cauchy data is the pair made by Dirichlet datum \(f\) and the normal derivative \(\partial_\nu u_{|\partial \Omega}\). The inverse problem consists into identifying the unknown obstacle \(D\), knowing the Cauchy data \(\{f, \partial_\nu u_{|\partial \Omega}\}\).
The authors prove that there is a duality between the range test (RT) and the no-response test (NRT) for the inverse boundary value problem. As an application, they show that either using the RT or NRT, we can reconstruct the obstacle \(D\) from the Cauchy data if the solution \(u\) does not have any analytic extension across \(\partial D\).Biharmonic problem with Dirichlet and Steklov-type boundary conditions in weighted spaceshttps://www.zbmath.org/1472.351322021-11-25T18:46:10.358925Z"Matevossian, H. A."https://www.zbmath.org/authors/?q=ai:matevossian.hovik-a|matevosyan.o-aSummary: The uniqueness of solutions of a biharmonic problem with Dirichlet and Steklov-type boundary conditions in the exterior of a compact set are studied under the assumption that the generalized solution of this problem has a finite Dirichlet integral with a weight \(| x |^a\). Depending on the parameter \(a\), uniqueness (non-uniqueness) theorems are proved and exact formulas for calculating the dimension of the solution space of this biharmonic problem are found.Asymptotic behavior of integral functionals for a two-parameter singularly perturbed nonlinear traction problemhttps://www.zbmath.org/1472.351872021-11-25T18:46:10.358925Z"Falconi, Riccardo"https://www.zbmath.org/authors/?q=ai:falconi.riccardo"Luzzini, Paolo"https://www.zbmath.org/authors/?q=ai:luzzini.paolo"Musolino, Paolo"https://www.zbmath.org/authors/?q=ai:musolino.paoloSummary: We consider a nonlinear traction boundary value problem for the Lamé equations in an unbounded periodically perforated domain. The edges lengths of the periodicity cell are proportional to a positive parameter \(\delta \), whereas the relative size of the holes is determined by a second positive parameter \(\varepsilon \). Under suitable assumptions on the nonlinearity, there exists a family of solutions \(\{ u (\varepsilon , \delta , \cdot ) \}_{( \varepsilon , \delta ) \in ]0, \varepsilon^{\prime}[ \times ]0, \delta^{\prime}[}\). We analyze the asymptotic behavior of two integral functionals associated to such a family of solutions when the perturbation parameter pair \((\varepsilon , \delta )\) is close to the degenerate value \((0, 0)\).Multiple results to some biharmonic problemshttps://www.zbmath.org/1472.352002021-11-25T18:46:10.358925Z"Tang, Xingdong"https://www.zbmath.org/authors/?q=ai:tang.xingdong"Zhang, Jihui"https://www.zbmath.org/authors/?q=ai:zhang.jihuiSummary: We study a nonlinear elliptic problem defined in a bounded domain involving biharmonic operator together with an asymptotically linear term. We establish at least three nontrivial solutions using the topological degree theory and the critical groups.Existence of the gauge for fractional Laplacian Schrödinger operatorshttps://www.zbmath.org/1472.354342021-11-25T18:46:10.358925Z"Frazier, Michael W."https://www.zbmath.org/authors/?q=ai:frazier.michael-w"Verbitsky, Igor E."https://www.zbmath.org/authors/?q=ai:verbitsky.igor-eSummary: Let \(\Omega\subseteq\mathbb{R}^n\) be an open set, where \(n\geq 2\). Suppose \(\omega\) is a locally finite Borel measure on \(\Omega\). For \(\alpha\in (0,2)\), define the fractional Laplacian \((-\Delta)^{\alpha/2}\) via the Fourier transform on \(\mathbb{R}^n\), and let \(G\) be the corresponding Green's operator of order \(\alpha\) on \(\Omega\). Define \(T(u)=G(u\omega)\). If \(\Vert T\Vert_{L^2(\omega)\rightarrow L^2(\omega)}<1\), we obtain a representation for the unique weak solution \(u\) in the homogeneous Sobolev space \(L^{\alpha/2,2}_0(\Omega)\) of
\[
(-\Delta)^{\alpha/2} u=u\omega+\nu\text{ on }\Omega,\quad u=0\text{ on }\Omega^c,
\]
for \(\nu\) in the dual Sobolev space \(L^{-\alpha/2,2}(\Omega)\). If \(\Omega\) is a bounded \(C^{1,1}\) domain, this representation yields matching exponential upper and lower pointwise estimates for the solution when \(\nu=\chi_{\Omega}\). These estimates are used to study the existence of a solution \(u_1\) (called the ``gauge'') of the integral equation \(u_1=1+G(u_1\omega)\) corresponding to the problem
\[
(-\Delta)^{\alpha/2}u=u\omega\text{ on }\Omega,\quad u\geq 0\text{ on }\Omega,\quad u=1\text{ on }\Omega^c.
\]
We show that if \(\Vert T\Vert<1\), then \(u_1\) always exists if \(0<\alpha<1\). For \(1\leq\alpha<2\), a solution exists if the norm of \(T\) is sufficiently small. We also show that the condition \(\Vert T\Vert <1\) does not imply the existence of a solution if \(1<\alpha<2\). The condition \(\Vert T\Vert\leq 1\) is necessary for the existence of \(u_1\) for all \(0<\alpha\leq 2\).State dependent Hamiltonian delay equations and Neumann one-formshttps://www.zbmath.org/1472.580072021-11-25T18:46:10.358925Z"Frauenfelder, Urs"https://www.zbmath.org/authors/?q=ai:frauenfelder.urs-adrianThe author proves critical point results for the action functionals involving Hamiltonian terms with a state dependent delay. The studied functionals are related to time dependent perturbations of the symplectic form. Results regarding Arnold type conjecture about lower bounds on the number of periodic orbits for Hamiltonian systems are given too.Some properties of the potential-to-ground state map in quantum mechanicshttps://www.zbmath.org/1472.813222021-11-25T18:46:10.358925Z"Garrigue, Louis"https://www.zbmath.org/authors/?q=ai:garrigue.louisThe author considers properties of the map from potential to the ground state in many-body quantum mechanics. External potentials \(v\in L^p + L^\infty\) and interaction potentials \(w \in L^p + L^\infty\) for \(p> \max(2d/3, 2)\) where \(d\) is the dimension of the underlying space are considered. The first result is that the space of binding potentials is path-connected. Then the author shows that the map from potentials to the ground state is locally weak-strong continuous and that its differential is compact. This implies that the Kohn-Sham inverse problem in Density Functional Theory is ill-posed on a bounded set.