Star-shaped Domains Research Articles

A Hybrid High-Order discretisation of the Brinkman problem robust in the Darcy and Stokes limits

In this work, we develop and analyse a novel Hybrid High-Order discretisation of the Brinkman problem. The method hinges on hybrid discrete velocity unknowns at faces and elements and on discontinuous pressures. Based on the discrete unknowns, we reconstruct inside each element a Stokes velocity one degree higher than face unknowns, and a Darcy velocity in the Raviart–Thomas–Nédélec space. These reconstructed velocities are respectively used to formulate the discrete versions of the Stokes and Darcy terms in the momentum equation, along with suitably designed penalty contributions. The proposed construction is tailored to yield optimal error estimates that are robust throughout the entire spectrum of local (Stokes- or Darcy-dominated) regimes, as identified by a dimensionless number which can be interpreted as a friction coefficient. The singular limit corresponding to the Darcy equation is also fully supported by the method. Numerical examples corroborate the theoretical results. This paper also contains two contributions whose interest goes beyond the specific method and application treated in this work: an investigation of the dependence of the constant in the second Korn inequality on star-shaped domains and its application to the study of the approximation properties of the strain projector in general Sobolev seminorms.

The harmonic virtual element method: stabilization and exponential convergence for the Laplace problem on polygonal domains

AbstractWe introduce the harmonic virtual element method (VEM) (harmonic VEM), a modification of the VEM (Beirão da Veiga et al. (2013) Basic principles of virtual element methods. Math. Models Methods Appl. Sci., 23, 199–214.) for the approximation of the two-dimensional Laplace equation using polygonal meshes. The main difference between the harmonic VEM and the VEM is that in the former method only boundary degrees of freedom are employed. Such degrees of freedom suffice for the construction of a proper energy projector on (piecewise harmonic) polynomial spaces. The harmonic VEM can also be regarded as an ‘$H^1$-conformisation’ of the Trefftz discontinuous Galerkin-finite element method (TDG-FEM) (Hiptmair et al. (2014) Approximation by harmonic polynomials in starshaped domains and exponential convergence of Trefftz hp-DGFEM. ESAIM Math. Model. Numer. Anal., 48, 727–752.). We address the stabilization of the proposed method and develop an hp version of harmonic VEM for the Laplace equation on polygonal domains. As in TDG-FEM, the asymptotic convergence rate of harmonic VEM is exponential and reaches order $\mathscr{O}(\exp (-b\sqrt [2]{N}))$, where $N$ is the number of degrees of freedom. This result overperforms its counterparts in the framework of hp FEM (Schwab, C. (1998)p- and hp-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics. Clarendon Press Oxford.) and hp VEM (Beirão da Veiga et al. (2018) Exponential convergence of the hp virtual element method with corner singularity. Numer. Math., 138, 581–613.), where the asymptotic rate of convergence is of order $\mathscr{O}(\exp(-b\sqrt [3]{N}))$.

Radial symmetry for p-harmonic functions in exterior and punctured domains

ABSTRACTWe prove symmetry for the p-capacitary potential satisfying under Serrin’s overdetermined condition Here is any bounded domain on which no a priori assumption is made, and denotes its boundary. Our result improves on a work of Garofalo and Sartori, where the same conclusion was obtained when is star-shaped. Our proof uses the maximum principle for an appropriate P-function, some integral identities, the isoperimetric inequality, and a Soap Bubble-type Theorem. We then treat the case , improving previous results present in the literature. Finally, with analogous tools, we give a new proof of symmetry for the interior overdetermined problem in a bounded star-shaped domain .

The Maslov index and the spectra of second order elliptic operators

We consider second order elliptic differential operators on a bounded Lipschitz domain Ω. Firstly, we establish a natural one-to-one correspondence between their self-adjoint extensions, with domains of definition containing in H1(Ω), and Lagrangian planes in H1/2(∂Ω)×H−1/2(∂Ω). Secondly, we derive a formula relating the spectral flow of the one-parameter families of such operators to the Maslov index, the topological invariant counting the signed number of conjugate points of paths of Lagrangian planes in H1/2(∂Ω)×H−1/2(∂Ω). Furthermore, we compute the Morse index, the number of negative eigenvalues, in terms of the Maslov index for several classes of the second order operators: the θ→-periodic Schrödinger operators on a period cell Q⊂Rn, the elliptic operators with Robin-type boundary conditions, and the abstract self-adjoint extensions of the Schrödinger operators on star-shaped domains. Our work is built on the techniques recently developed by B. Booß-Bavnbek, K. Furutani, and C. Zhu, and extends the scope of validity of their spectral flow formula by incorporating the self-adjoint extensions of the second order operators with domains in the first order Sobolev space H1(Ω). In addition, we generalize the results concerning relations between the Maslov and Morse indices quite recently obtained by G. Cox, J. Deng, C. Jones, J. Marzuola, A. Sukhtayev and the authors. Finally, we describe and study a link between the theory of abstract boundary triples and the Lagrangian description of self-adjoint extensions of abstract symmetric operators.

A scaled boundary isogeometric formulation for the elasto-plastic analysis of solids in boundary representation

This contribution deals with the nonlinear analysis of boundary represented solids with elasto-plastic material behavior based on the so-called scaled boundary isogeometric formulation (SB-IGA). The proposed approach combines the features of the scaled boundary finite element method and isogeometric analysis. Based on the original boundary representation of the CAD model, a formulation is provided where the geometrical description of the boundary is sufficient to define the entire surface. The domain is parameterized by a radial scaling parameter emanating from a scaling center and a parameter in circumferential direction along the boundary. Non star-shaped domains are tackled by standard sub-structuring. Here, conforming discretizations are considered for the two-dimensional case. According to the isogeometric paradigm, NURBS basis functions are employed for the approximation of the solution. The displacement response is derived based on a multiplicative decomposition of the approximation in circumferential and radial scaling direction. The boundary value problem is solved with the Galerkin method. The Newton–Raphson iterative scheme is employed to obtain the nonlinear response. Several benchmark tests demonstrate the accuracy and computational efficiency of the formulation.

Lower bounds for blow-up time in nonlocal parabolic problem under Robin boundary conditions

ABSTRACTThis paper deals with the blow-up phenomenon for a nonlocal reaction diffusion equation with time-dependent coefficient in a bounded star-shaped domain under Robin boundary condition. Using the auxiliary function method and differential inequality technique, lower bounds for the blow-up time can be obtained if the blow-up occurs in finite time. Some examples are presented to illustrate applications of our results.

General Solution of the Inhomogeneous Div-Curl System and Consequences

We consider the inhomogeneous div-curl system (i.e. to find a vector field with prescribed div and curl) in a bounded star-shaped domain in $$\mathbb {R}^3$$ . An explicit general solution is given in terms of classical integral operators, completing previously known results obtained under restrictive conditions. This solution allows us to solve questions related to the quaternionic main Vekua equation $$DW=(Df/f)\overline{W}$$ in $$\mathbb {R}^3$$ , such as finding the vector part when the scalar part is known. In addition, using the general solution to the div-curl system and the known existence of the solution of the inhomogeneous conductivity equation, we prove the existence of solutions of the inhomogeneous double curl equation, and give an explicit solution for the case of static Maxwell’s equations with only variable permeability.

Quaternionic Functions and Their Applications in a Viscous Fluid Flow

We used a quaternion function method for the Moisil–Theodoresco system (MTS). Solutions of the MTS are (left-) regular quaternion functions $$f(\mathbf{r})=f_0(\mathbf{r})+\mathbf{f}(\mathbf{r})= f_0(x,y,z)+\mathbf{i}f_x(x,y,z)+\mathbf{j}f_y(x,y,z)+\mathbf{r}f_z(x,y,z)$$ of a reduced quaternion variable $$\mathbf{r}=\mathbf{i}x + \mathbf{j}y + \mathbf{k}z$$ . Here we present the quaternion three-dimensional representation of a general solution of the Stokes system for the slow flow of a viscous fluid in star-shaped domain. It is shown that in particular cases of plane and axially symmetric flows this representation goes into the representations by means of analytical and generalized analytical functions of complex variables. As applications the main problems the Stokes flow in a ball are solved.

The direct method of lines for elliptic problems in star-shaped domains

In this paper, we generalize the direct method of lines for elliptic problems in star-shaped domains. We assume that the boundary of the star-shaped domain is a closed Lipschitz curve that can be parameterized by the angular variable, so that an appropriate transformation of coordinates can be introduced. Then the elliptic problem is reduced to a variational–differential problem on a semi-infinite strip in the new coordinates. We discretize the reduced problem with respect to the angular variable and obtain a semi-discrete approximation. Then a direct method is adopted to solve the semi-discrete problem analytically. Finally, the optimal error estimate of the semi-discrete approximation is given and several numerical examples are presented to show that our method is feasible and effective for a wide range of elliptic problems.

On a class of (p; q)-Laplacian problems involving the critical Sobolev-Hardy exponents in starshaped domain

Abstract Let Ω ⊂ ℝn be a bounded starshaped domain and consider the (p; q)-Laplacian problem -∆pu - ∆pu = λ(x)|u|p*-2u + μ|u|r-2u where μ is a positive parameter, 1 < q ≤ p < n, r ≥ p* and is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the (p; q)-Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.

Large solutions for cooperative logistic systems: existence and uniqueness in star-shaped domains

Abstract We extend Theorem 1.1 of [J. Math. Anal. Appl. 435 (2016), 1738-1752] to show the uniqueness of large solutions for the system of (1) in star-shaped domains. This result is due to the maximum principle for cooperative systems of [J. López-Gómez and M. Molina-Meyer, Diff. Int. Eq. 7 (1994), 383-398], which allows us to establish the uniqueness without invoking to the blow-up rates of the solutions.

Local null controllability of one-phase Stefan problems in 2D star-shaped domains

In this work, we are concerned with the null controllability of one-phase Stefan problems in star-shaped domains in $$\mathbb {R}^2$$ . We prove that, for fixed $$T>0$$ and sufficient small initial data, there exist controls that drive the state to zero at time $$t=T$$ . Our approach relies on a null controllability result for parabolic systems in non-cylindrical domains, a complete analysis of the regularity of the controlled solution near the free boundary and a fixed-point argument.

Hardy inequality for star-shaped domains

Hardy inequality for star-shaped domains

Sharp Hardy Inequalities in Star-shaped Domains with Double Singular Kernels

Improved Hardy inequality with double singular kernel in bounded, star-shaped domains \(\Omega \subset R^n\) is proved. The case of singularity at an interior point and on the boundary of the domain is considered. The Hardy constant is optimal and the inequality is sharp due to the additional term.

Blowup for fractional NLS

We consider fractional NLS with focusing power-type nonlinearityi∂tu=(−Δ)su−|u|2σu,(t,x)∈R×RN, where 1/2<s<1 and 0<σ<∞ for s⩾N/2 and 0<σ⩽2s/(N−2s) for s<N/2. We prove a general criterion for blowup of radial solutions in RN with N⩾2 for L2-supercritical and L2-critical powers σ⩾2s/N. In addition, we study the case of fractional NLS posed on a bounded star-shaped domain Ω⊂RN in any dimension N⩾1 and subject to exterior Dirichlet conditions. In this setting, we prove a general blowup result without imposing any symmetry assumption on u(t,x).For the blowup proof in RN, we derive a localized virial estimate for fractional NLS in RN, which uses Balakrishnan's formula for the fractional Laplacian (−Δ)s from semigroup theory. In the setting of bounded domains, we use a Pohozaev-type estimate for the fractional Laplacian to prove blowup.

The Morse and Maslov indices for multidimensional Schrödinger operators with matrix-valued potentials

We study the Schrodinger operator L = −∆ +V on a star-shaped domain Ω in Rd with Lipschitz boundary ∂Ω. The operator is equipped with quite general Dirichletor Robin-type boundary conditions induced by operators between H1/2(∂Ω) and H−1/2(∂Ω), and the potential takes values in the set of symmetric N ×N matrices. By shrinking the domain and rescaling the operator we obtain a path in the Fredholm–Lagrangian Grassmannian of the subspace of H1/2(∂Ω)×H−1/2(∂Ω) corresponding to the given boundary condition. The path is formed by computing the Dirichlet and Neumann traces of weak solutions to the rescaled eigenvalue equation. We prove a formula relating the number of negative eigenvalues of L (the Morse index), the signed crossings of the path (the Maslov index), the number of negative eigenvalues of the potential matrix evaluated at the center of the domain, and the number of negative eigenvalues of a bilinear form related to the boundary operator.

Optimal polynomial admissible meshes on some classes of compact subsets of [formula omitted

We show that any compact subset of Rd which is the closure of a bounded star-shaped Lipschitz domain Ω, such that ∁Ω has positive reach in the sense of Federer, admits an optimal AM (admissible mesh), that is a sequence of polynomial norming sets with optimal cardinality. This extends a recent result of A. Kroó on C2 star-shaped domains.Moreover, we prove constructively the existence of an optimal AM for any K:=Ω¯⊂Rd where Ω is a bounded C1,1 domain. This is done by a particular multivariate sharp version of the Bernstein Inequality via the distance function.

Partial Differential Equation for Evolution of Star-Shaped Reachability Domains of Differential Inclusions

The problem of reachability for differential inclusions is an active topic in the recent control theory. Its solution provides an insight into the dynamics of an investigated system and also enables one to design synthesizing control strategies under a given optimality criterion. The primary results on reachability were mostly applicable to convex sets, whose dynamics is described through that of their support functions. Those results were further extended to the viability problem and some types of nonlinear systems. However, non-convex sets can arise even in simple bilinear systems. Hence, the issue of nonconvexity in reachability problems requires a more detailed investigation. The present article follows an alternative approach for this cause. It deals with star-shaped reachability sets, describing the evolution of these sets in terms of radial (Minkowski gauge) functions. The derived partial differential equation is then modified to cope with additional state constraints due to on-line measurement observations. Finally, the last section is on designing optimal closed-loop control strategies using radial functions.

Smooth Billiards with a Large Weyl Remainder

We prove an estimate from below for the remainder in Weyl's law for smooth star-shaped planar domains admitting an appropriate one-parameter family of periodic billiard trajectories. Examples include ellipses and smooth domains of constant width. Our results confirm a prediction of P. Sarnak who proved a similar statement for surfaces without boundary. The proof is based on the analysis of the singularity in the wave trace, occurring at time T>0T>0 equal to the length of the closed trajectories belonging to the family. Such families of periodic orbits produce large singularities in the wave trace, which in turn imply estimates from below for the absolute value of the Weyl remainder averaged over a spectral window. We also obtain lower bounds for the error term in higher dimensions. In this case, the main contribution to the Weyl remainder typically comes from the singularity at zero of the wave trace. However, for certain domains, such as the Euclidean ball, the dimension of the family of periodic trajectories is large enough to dominate the contribution of the singularity at zero.

A Transformation Approach in Shape Optimization: Existence and Regularity Results

In this article, we consider a model shape optimization problem. The state variable solves an elliptic equation on a star-shaped domain, where the radius is given via a control function. First, we reformulate the problem on a fixed reference domain, where we focus on the regularity needed to ensure the existence of an optimal solution. Second, we introduce the Lagrangian and use it to show that the optimal solution possesses a higher regularity, which allows for the explicit computation of the derivative of the reduced cost functional as a boundary integral. We finish the article with some second-order optimality conditions.