Recent zbMATH articles in MSC 31https://www.zbmath.org/atom/cc/312021-04-16T16:22:00+00:00WerkzeugSpectral characteristics of the self-balanced stress fields.https://www.zbmath.org/1456.740052021-04-16T16:22:00+00:00"Guzev, M. A."https://www.zbmath.org/authors/?q=ai:guzev.mikhail-a|guzev.mickhail-aSummary: We investigate a class of self-balanced stress fields which is parameterized by a stress function. The fuction is considered to be an element of the spectrum of the biharmonic operator. For different types of boundary conditions we constructed the spectral characteristics of the operator.Sinc-Galerkin solution to the clamped plate eigenvalue problem.https://www.zbmath.org/1456.651502021-04-16T16:22:00+00:00"El-Gamel, Mohamed"https://www.zbmath.org/authors/?q=ai:el-gamel.mohamed"Mohsen, Adel"https://www.zbmath.org/authors/?q=ai:mohsen.adel-a-k"Abdrabou, Amgad"https://www.zbmath.org/authors/?q=ai:abdrabou.amgadSummary: We propose an accurate and computationally efficient numerical technique for solving the biharmonic eigenvalue problem. The technique is based on the sinc-Galerkin approximation method to solve the clamped plate problem. Numerical experiments for plates with various aspect ratios are reported, and comparisons are made with other methods in literature. The calculated results accord well with those published earlier, which proves the accuracy and validity of the proposed method.Minimal energy point systems on the unit circle and the real line.https://www.zbmath.org/1456.310022021-04-16T16:22:00+00:00"Gaál, Marcell"https://www.zbmath.org/authors/?q=ai:gaal.marcell"Nagy, Béla"https://www.zbmath.org/authors/?q=ai:nagy.bela.1"Nagy-Csiha, Zsuzsanna"https://www.zbmath.org/authors/?q=ai:nagy-csiha.zsuzsanna"Révész, Szilárd Gy."https://www.zbmath.org/authors/?q=ai:revesz.szilard-gyorgyThe two-grid discretization of Ciarlet-Raviart mixed method for biharmonic eigenvalue problems.https://www.zbmath.org/1456.651782021-04-16T16:22:00+00:00"Zhang, Yu"https://www.zbmath.org/authors/?q=ai:zhang.yu.3"Bi, Hai"https://www.zbmath.org/authors/?q=ai:bi.hai"Yang, Yidu"https://www.zbmath.org/authors/?q=ai:yang.yiduSummary: In this paper, for biharmonic eigenvalue problems with clamped boundary condition in \(\mathbb{R}^n\) which include plate vibration problem and plate buckling problem, we primarily study the two-grid discretization based on the shifted-inverse iteration of Ciarlet-Raviart mixed method. With our scheme, the solution of a biharmonic eigenvalue problem on a fine mesh \(\pi_h\) can be reduced to the solution of an eigenvalue problem on a coarser mesh \(\pi_H\) and the solution of a linear algebraic system on the fine mesh \(\pi_h\). With a new argument which is not covered by existing work, we prove that the resulting solution still maintains an asymptotically optimal accuracy when \(H > h \geq \mathcal{O}(H^2)\). The surprising numerical results show the efficiency of our scheme.Arsove-Huber theorem in higher dimensions.https://www.zbmath.org/1456.310052021-04-16T16:22:00+00:00"Ma, Shiguang"https://www.zbmath.org/authors/?q=ai:ma.shiguang"Qing, Jie"https://www.zbmath.org/authors/?q=ai:qing.jieThe aim is to extend the Arsove-Huber theory of surfaces to higher dimensions. A basic tool is the \(n\)-Laplace equation \[\text{div}(|\nabla u|^{n-2}\nabla u)= 0\] in exactly \(n\) dimensions (the so-called borderline case). The Brezis-Merle inequality (a refined version of Trudinger's inequality) is applied in \(n\) dimensions. An ingredient is the Wolff potential.
The theory is applied for hypersurfaces in a hyperbolic space, having nonnegative Ricci curvature. Even some unpublished results are announced.
For the entire collection see [Zbl 1446.58001].
Reviewer: Peter Lindqvist (Trondheim)Balayage of measures with respect to (sub-)harmonic functions.https://www.zbmath.org/1456.310042021-04-16T16:22:00+00:00"Khabibullin, B. N."https://www.zbmath.org/authors/?q=ai:khabibullin.b-nSummary: We investigate some properties of balayage, or, sweeping (out), of measures with respect to subclasses of subharmonic functions. The following issues are considered: relationships between balayage of measures with respect to classes of harmonic or subharmonic functions and balayage of measures with respect to significantly smaller classes of specific classes of functions; integration of measures and balayage of measures; sensitivity of balayage of measures to polar sets, etc.The Liouville theorem for \(p\)-harmonic functions and quasiminimizers with finite energy.https://www.zbmath.org/1456.350552021-04-16T16:22:00+00:00"Björn, Anders"https://www.zbmath.org/authors/?q=ai:bjorn.anders"Björn, Jana"https://www.zbmath.org/authors/?q=ai:bjorn.jana"Shanmugalingam, Nageswari"https://www.zbmath.org/authors/?q=ai:shanmugalingam.nageswariSummary: We show that, under certain geometric conditions, there are no nonconstant quasiminimizers with finite \(p\)th power energy in a (not necessarily complete) metric measure space equipped with a globally doubling measure supporting a global \(p\)-Poincaré inequality. The geometric conditions are that either (a) the measure has a sufficiently strong volume growth at infinity, or (b) the metric space is annularly quasiconvex (or its discrete version, annularly chainable) around some point in the space. Moreover, on the weighted real line \(\mathbf{R}\), we characterize all locally doubling measures, supporting a local \(p\)-Poincaré inequality, for which there exist nonconstant quasiminimizers of finite \(p\)-energy, and show that a quasiminimizer is of finite \(p\)-energy if and only if it is bounded. As \(p\)-harmonic functions are quasiminimizers they are covered by these results.A Riemann jump problem for biharmonic functions in fractal domains.https://www.zbmath.org/1456.310032021-04-16T16:22:00+00:00"Abreu Blaya, R."https://www.zbmath.org/authors/?q=ai:abreu-blaya.ricardoSummary: Biharmonic functions are the solutions of the fourth order partial differential equation \(\Delta \Delta \omega =0\). The purpose of this paper is to solve a kind of Riemann boundary value problem for biharmonic functions assuming higher order Lipschitz boundary data. We approach this problem making use of generalized Teodorescu transforms for obtaining the explicit expression of its solution in a Jordan domain \(\Omega \subset \mathbb{R}^2\) with fractal boundary.Hyperbolic harmonic functions and hyperbolic Brownian motion.https://www.zbmath.org/1456.601942021-04-16T16:22:00+00:00"Eriksson, Sirkka-Liisa"https://www.zbmath.org/authors/?q=ai:eriksson.sirkka-liisa"Kaarakka, Terhi"https://www.zbmath.org/authors/?q=ai:kaarakka.terhiSummary: We study harmonic functions with respect to the Riemannian metric
\[ds^2=\frac{dx_1^2+\cdots +dx_n^2}{x_n^{\frac{2\alpha}{n-2}}}\] in the upper half space \(\mathbb{R}_+^n=\{(x_1,\dots,x_n) \in \mathbb{R}^n :x_n>0\}\). They are called \(\alpha\)-hyperbolic harmonic. An important result is that a function \(f\) is \(\alpha\)-hyperbolic harmonic íf and only if the function \(g(x) =x_n^{-\frac{2-n+\alpha}{2}}f(x)\) is the eigenfunction of the hyperbolic Laplace operator \(\triangle_h=x_n^2\triangle -(n-2) x_n\frac{\partial}{\partial x_n}\) corresponding to the eigenvalue \(\frac{1}{4} ((\alpha+1)^2-(n-1)^2)=0\). This means that in case \(\alpha =n-2\), the \(n-2\)-hyperbolic harmonic functions are harmonic with respect to the hyperbolic metric of the Poincaré upper half-space. We are presenting some connections of \(\alpha\)-hyperbolic functions to the generalized hyperbolic Brownian motion. These results are similar as in case of harmonic functions with respect to usual Laplace and Brownian motion.The quasisuperminimizing constant for the minimum of two quasisuperminimizers in \(R^n\).https://www.zbmath.org/1456.310102021-04-16T16:22:00+00:00"Björn, Anders"https://www.zbmath.org/authors/?q=ai:bjorn.anders"Björn, Jana"https://www.zbmath.org/authors/?q=ai:bjorn.jana"Mirumbe, Ismail"https://www.zbmath.org/authors/?q=ai:mirumbe.ismailQuasiminimizers were introduced in [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 79--107 (1984; Zbl 0541.49008)] by \textit{M. Giaquinta} and \textit{E. Giusti}, who noticed that they belong to the De Giorgi class. A function \(u\) in the Sobolev space \(W^{1,p}_{\text{loc}}(\Omega)\) is a \(Q\)-quasiminimizer in the domain \(\Omega\) in \(\mathbb{R}^n\) if
\[
\int_{\operatorname{supp}\{\phi\}}|\nabla u|^pdx\le Q \int_{\operatorname{supp}\{\phi\}}|\nabla u+\nabla\phi|^pdx
\]
for all \(\phi\in W^{1,p}_0(\Omega)\). Here \(1<p<\infty\) and \(Q\) is a constant. \(Q\)-quasisuperminimizers are defined analogously, but under the restriction \(\phi\ge 0\) for the test functions.
If \(u_1\) is a \(Q_1\)-quasisuperminimizers and \(u_2\) a \(Q_2\)-quasisuperminimizer, then \(\max\{u_1,u_2\}\) is a \(Q\)-quasisuperminimizer for some \(Q\). The sharpness of a formula for a nearly optimal \(Q\) is the topic of the present work. Explicit functions like \(|x|^\alpha(\log^+(|x|))^\beta\) are utilized. In the important borderline case \(p=n\ge 2\) the following example is constructed in the ring domain \(1/e<|x|<1\). Let \(Q_2\ge Q_1\ge 1\) be given. Two functions \(u_1\) and \(u_2\) are exhibited: \(u_j\) is a \(Q_j\)-quasisuperminimizer \((j= 1,2)\), but \(\max\{u_1,u_2\}\) is not a \(Q_2\)-quasisuperminimizer.
Some numerical tables for the constants \(Q\) are constructed for radial functions.
Reviewer: Peter Lindqvist (Trondheim)A concept of weak Riesz energy with application to condensers with touching plates.https://www.zbmath.org/1456.310092021-04-16T16:22:00+00:00"Zorii, Natalia"https://www.zbmath.org/authors/?q=ai:zorii.nataliaSummary: We proceed further with the study of minimum weak Riesz energy problems for condensers with touching plates, initiated jointly with \textit{B. Fuglede} [Potential Anal. 51, No. 2, 197--217 (2019; Zbl 1432.31006)]. Having now added to the analysis constraint and external source of energy, we obtain a Gauss type problem, but with weak energy involved. We establish sufficient and/or necessary conditions for the existence of solutions to the problem and describe their potentials. Treating the solution as a function of the condenser and the constraint, we prove its continuity relative to the vague topology and the topologies determined by the weak and standard energy norms. We show that the criteria for the solvability thus obtained fail in general once the problem is reformulated in the setting of standard energy, thereby justifying an advantage of weak energy when dealing with condensers with touching plates.On simple eigenvalues of the fractional Laplacian under removal of small fractional capacity sets.https://www.zbmath.org/1456.310072021-04-16T16:22:00+00:00"Abatangelo, Laura"https://www.zbmath.org/authors/?q=ai:abatangelo.laura"Felli, Veronica"https://www.zbmath.org/authors/?q=ai:felli.veronica"Noris, Benedetta"https://www.zbmath.org/authors/?q=ai:noris.benedettaAuthors' abstract: We consider the eigenvalue problem for the restricted fractional Laplacian in a bounded domain with homogeneous Dirichlet boundary conditions. We introduce the notion of fractional capacity for compact subsets, with the property that the eigenvalues are not affected by the removal of zero fractional capacity sets. Given a simple eigenvalue, we remove from the domain a family of compact sets which are concentrating to a set of zero fractional capacity and we detect the asymptotic expansion of the eigenvalue variation; this expansion depends on the eigenfunction associated to the limit eigenvalue. Finally, we study the case in which the family of compact sets is concentrating to a point.
Reviewer: Dian K. Palagachev (Bari)Various concepts of Riesz energy of measures and application to condensers with touching plates.https://www.zbmath.org/1456.310082021-04-16T16:22:00+00:00"Fuglede, Bent"https://www.zbmath.org/authors/?q=ai:fuglede.bent"Zorii, Natalia"https://www.zbmath.org/authors/?q=ai:zorii.nataliaLet \(n\geq 3\) and \(0<\alpha \leq 2\). The \(\alpha \)-Riesz potential of a signed Radon measure \(\mu \) on \(\mathbb{R}^{n}\) is defined by \(\kappa_{\alpha }\mu (x)=\int \left\vert x-y\right\vert ^{\alpha -n}d\mu (y)\), and the usual \(\alpha \)-Riesz energy is given by \(E_{\alpha }(\mu )=\int \kappa_{\alpha }\mu ~d\mu \). However, this notion of energy is unsuitable for studying minimum energy problems for condensers with touching plates. This motivates the authors to develop further the theory of weak \(\alpha \)-Riesz energy, which they recently introduced in [Potential Anal. 51, No. 2, 197--217 (2019; Zbl 1432.31006)]. This is defined by \(\int (\kappa _{\alpha /2}\mu )^{2}dm\), where \(m\) denotes the Lebesgue measure on \(\mathbb{R}^{n}\), and coincides with \(E_{\alpha}(\mu )\) when \(\mu \) is a positive measure. The authors investigate minimum weak \(\alpha \)-Riesz energy problems with external fields (in both the constrained and unconstrained settings) for condensers with plates that may touch. They also describe the relationship between minimum weak \(\alpha \)-Riesz energy problems over signed measures associated with generalized condensers \((A_{1},A_{2})\) and minimum \(\alpha \)-Green energy problems over positive measures carried by \(A_{1}\). They succeed in recovering and improving results that were announced in [\textit{P. D. Dragnev} et al., Potential Anal. 44, No. 3, 543--577 (2016; Zbl 1338.31008)], but which relied on a lemma that turned out to be erroneous.
Reviewer: Stephen J. Gardiner (Dublin)Hereditary circularity for energy minimal diffeomorphisms.https://www.zbmath.org/1456.300432021-04-16T16:22:00+00:00"Koh, Ngin-Tee"https://www.zbmath.org/authors/?q=ai:koh.ngin-teeSummary: We reveal some geometric properties of energy minimal diffeomorphisms defined on an annulus, whose existence was established in works by
\textit{T. Iwaniec} et al. [Invent. Math. 186, No. 3, 667--707 (2011; Zbl 1255.30031); J. Am. Math. Soc. 24, No. 2, 345--373 (2011; Zbl 1214.31001)]
and \textit{D. Kalaj} [Calc. Var. Partial Differ. Equ. 51, No. 1--2, 465--494 (2014; Zbl 1296.30052)].On a new subclass of harmonic univalent functions.https://www.zbmath.org/1456.310012021-04-16T16:22:00+00:00"Bayram, H."https://www.zbmath.org/authors/?q=ai:bayram.hasan|bayram.husamettin"Yalçin, S."https://www.zbmath.org/authors/?q=ai:yalcin.seth|yalcin.senay|yalcin.sibel|yalcin.serife-nur|yalcin.soydanSummary: In the acquaint article, we scrutinize some fundamental attribute of a subclass of harmonic univalent functions defined by a new alteration. Like these, coefficient disparities, distortion bounds, convolutions, convex combinations and extreme points.On \(C^m\)-reflection of harmonic functions and \(C^m\)-approximation by harmonic polynomials.https://www.zbmath.org/1456.310062021-04-16T16:22:00+00:00"Paramonov, Petr V."https://www.zbmath.org/authors/?q=ai:paramonov.peter-v"Fedorovskiy, Konstantin Yu."https://www.zbmath.org/authors/?q=ai:fedorovskiy.konstantin-yuThe paper deals with sharp \(C^m\) continuity conditions for harmonic reflection operators of functions over the boundary of simple Carathéodory domains in \(\mathbb{R}^N\). A simple Carathéodory domain \(D \subset \mathbb{R}^N\), \(N\geq 2\), is a nonempty bounded domain sucht that the set \(\Omega \equiv \mathbb{R}^N \setminus \overline{D}\) is a domain, \(\partial D=\partial \Omega\) and, if \(N\geq 3\), both \(D\) and \(\Omega\) are regular with respect to the Dirichlet problem for harmonic functions. The harmonic reflection operator \(R_D\) is the map which takes a sufficiently regular harmonic function \(f\) in \(D\) to the solution \(g\) of the exterior Dirichlet problem for the Laplace operator in the complement of the closure of \(D\) and with Dirichlet data the trace of \(f\) on \(\partial D\). The authors establish sharp conditions on a Carathéodory domain ensuring that the operator \(R_D\) preserves the \(C^m\) continuity, \(m\in(0,1)\), of functions. This is done by means of a new criterion for the \(C^m\) continuity of the Poisson operator, i.e., the operator that maps a smooth function \(\phi\) on \(\partial D\) to the solution \(f\) of the Dirichlet problem with boundary data \(\phi\) on \(\partial D\). Finally, the authors deduce sufficient conditions for the \(C^m\) approximation of functions by harmonic polynomials on \(\partial D\).
Reviewer: Paolo Musolino (Padova)On Hilbert and Riemann problems for generalized analytic functions and applications.https://www.zbmath.org/1456.300872021-04-16T16:22:00+00:00"Ryazanov, Vladimir"https://www.zbmath.org/authors/?q=ai:ryazanov.vladimir-i|ryazanov.vladimir-vSummary: The research of the Dirichlet problem with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin. The present paper is devoted to various theorems on the existence of nonclassical solutions of the Hilbert and Riemann boundary value problems with arbitrary measurable data for generalized analytic functions by Vekua and the corresponding applications to the Neumann and Poincaré problems for generalized harmonic functions. Our approach is based on the geometric (theoretic-functional) interpretation of boundary values in comparison with the classical operator approach in PDE. First of all, here it is proved theorems on the existence of solutions to the Hilbert boundary value problem with arbitrary measurable data for generalized analytic functions in arbitrary Jordan domains with rectifiable boundaries in terms of the natural parameter and angular (nontangential) limits, moreover, in arbitrary Jordan domains in terms of harmonic measure and principal asymptotic values. Moreover, it is established the existence theorems on solutions for the appropriate boundary value problems of Hilbert and Riemann with arbitrary measurable data along the so-called Bagemihl-Seidel systems of Jordan arcs terminating at the boundary in arbitrary domains whose boundaries consist of finite collections of rectifiable Jordan curves. On this basis, it is established the corresponding existence theorems for the Poincaré boundary value problem on the directional derivatives and, in particular, for the Neumann problem with arbitrary measurable data to the Poisson equations.A variable exponent Sobolev theorem for fractional integrals on quasimetric measure spaces.https://www.zbmath.org/1456.460362021-04-16T16:22:00+00:00"Samko, Stefan"https://www.zbmath.org/authors/?q=ai:samko.stefan-gSummary: We show that the fractional operator \(I^{\alpha(\cdot)}\) of variable order on a bounded open set \(\Omega\) in a quasimetric measure space \((X,d\mu)\) with the growth condition the measure \(\mu\), is bounded from the variable exponent Lebesgue space \(L^{p(\cdot)}(\Omega)\) into \(L^{q(\cdot)}(\Omega)\) in the case \(\inf_{x\in\Omega}[n(x)-\alpha(x)p(x)] > 0\), where \(\frac{1}{q(x)}=\frac{1}{p(x)}-\frac{\alpha(x)}{n(x)}\) and \(n(x)\) comes from the growth condition under the log-continuity condition on \(p(x)\).A strong form of Plessner's theorem.https://www.zbmath.org/1456.300232021-04-16T16:22:00+00:00"Gardiner, Stephen J."https://www.zbmath.org/authors/?q=ai:gardiner.stephen-j"Manolaki, Myrto"https://www.zbmath.org/authors/?q=ai:manolaki.myrtoThe authors strengthen classical results on boundary behavior of holomorphic and harmonic functions. Denote \(\mathbb D=\{z\in\mathbb C:|z|<1\}\), \(\mathbb T=\partial\mathbb D\), \(C_{w,r}=\{z\in\mathbb C:|z-w|=r\}\), \(w\in\mathbb C\), \(r>0\), and \(\lambda_n\) the Lebesgue measure on \(\mathbb R^n\), \(n\geq1\). A Stolz angle at \(\zeta\in\mathbb T\) is an open triangular subset of \(\mathbb D\) that has a vertex at \(\zeta\) and is symmetric about the the diameter of \(\mathbb D\) through \(\zeta\). The main result of the paper is presented in the following theorem.
Theorem 1. Let \(f\) be a holomorphic function on \(\mathbb D\). Then, for \(\lambda_1\)-almost every point \(\zeta\in\mathbb T\), either \(f\) has a finite nontangential limit at \(\zeta\), or for every Stolz angle \(S\) at \(\zeta\), \[\int_{S\cap f^{-1}(C_{w,r})}|f'(z)||dz|=\infty\] for \(\lambda_1\)-almost every \((w,r)\in\mathbb C\times(0,\infty)\).
Theorem 1 is generalized to meromorphic functions. The authors prove an analogue of Theorem 1 for harmonic functions on the halfspace \(\mathbb H=\{(x_1,\dots,x_N)\in\mathbb R^N:x_N>0\}\), \(N\geq2\). A Stolz domain at \(y\in\partial\mathbb H\) is a truncated cone in \(\mathbb H\) that meets \(\partial\mathbb H\) at its vertex \(y\) and with its axis normal to \(\partial\mathbb H\).
Theorem 4. Let \(h\) be a harmonic function on \(\mathbb H\). Then, for \(\lambda_{N-1}\)-almost every point \(y\in\partial\mathbb H\), either \(h\) has a finite nontangential limit at \(y\), or for every Stolz domain \(S\) at \(y\), \[\int_{S\cap h^{-1}(\{t\})}x_N^{2-N}\|\nabla h(x)\|d\sigma(x)=\infty\] for \(\lambda_1\)-almost every \(t\in\mathbb R\), where \(\sigma\) is the surface area measure on level sets of (nonconstant) harmonic functions.
Reviewer: Dmitri V. Prokhorov (Saratov)