Recent zbMATH articles in MSC 31Ahttps://www.zbmath.org/atom/cc/31A2021-04-16T16:22:00+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-gyorgyA 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)The 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.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.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 \(p(x)\).