Ellipticity Condition Research Articles

Nonlinear strain gradient and micromorphic one-dimensional elastic continua: Comparison through strong ellipticity conditions

We discuss the strong ellipticity (SE) conditions for strain gradient and micromorphic continua considering them as an enhancement of a simple nonlinearly elastic material called in the following primary material. Recently both models are widely used for description of material behavior of beam-lattice metamaterials which may possess various types of material instabilities. We analyze how a possible loss of SE results in the behavior of enhanced models. We shown that SE conditions for a micromorphic medium is more restrictive than for its gradient counterpart. On the other hand we see that a violation of SE for a primary material affects solutions within enhanced models even if the SE conditions are fulfilled for them.

Lp-regularity theory for semilinear stochastic partial differential equations with multiplicative white noise

We establish the Lp-regularity theory for a semilinear stochastic partial differential equation with multiplicative white noise:du=(aijuxixj+biuxi+cu+b¯i|u|λuxi)dt+σk(u)dwtk,(ω,t,x)∈Ω×(0,∞)×Rd;u(0,⋅)=u0, where λ>0, the set {wtk,k=1,2,…} is a set of one-dimensional independent Wiener processes, and the function u0=u0(ω,x) is nonnegative random initial data. The coefficients aij,bi,c depend on (ω,t,x), and b¯i depends on (ω,t,x1,…,xi−1,xi+1,…,xd). The coefficients aij,bi,c,b¯i are uniformly bounded and twice continuously differentiable. The leading coefficient a satisfies the ellipticity condition. Depending on the diffusion coefficient σk(u)=σk(ω,t,x,u), we consider two cases:(i)λ∈(0,∞) and σk(u) has a Lipschitz continuity and linear growth in u.(ii)λ,λ0∈(0,1/d) and σk(ω,t,x,u)=μk(ω,t,x)|u|1+λ0 with |∑k|μk(ω,t,⋅)|2|C(Rd)<∞ for all (ω,t)∈Ω×[0,∞). We obtain the existence, uniqueness, Lp regularity, and Hölder regularity of the solution. It should be noted that each case has different regularity results. For example, in the case of (i), for ε>0u∈Ct,x1/2−ε,1−ε([0,T]×Rd)∀T<∞, almost surely. On the other hand, in the case of (ii), if λ,λ0∈(0,1/d), for ε>0u∈Ct,x1−(λd)∨(λ0d)2−ε,1−(λd)∨(λ0d)−ε([0,T]×Rd)∀T<∞ almost surely. It should be noted that λ can be any positive number and that the solution regularity is independent of the nonlinear terms in case (i). In case (ii), however, λ and λ0 should satisfy λ,λ0∈(0,1/d), and the regularities of the solution are affected by λ,λ0 and d. This difference is due to the need to use a different proof method for the super-linear diffusion coefficient σk(u).

A regularity theory for stochastic generalized Burgers’ equation driven by a multiplicative space-time white noise

We introduce the uniqueness, existence, $$L_p$$ -regularity, and maximal Hölder regularity of the solution to semilinear stochastic partial differential equation driven by a multiplicative space-time white noise: $$\begin{aligned} u_t = au_{xx} + bu_{x} + cu + {{\bar{b}}}|u|^\lambda u_{x} + \sigma (u)\dot{W},\quad (t,x)\in (0,\infty )\times {\mathbb {R}}; \quad u(0,\cdot ) = u_0, \end{aligned}$$ where $$\lambda > 0$$ . The function $$\sigma (u)$$ is either bounded Lipschitz or super-linear in u. The noise $$\dot{W}$$ is a space-time white noise. The coefficients a, b, c depend on $$(\omega ,t,x)$$ , and $${{\bar{b}}}$$ depends on $$(\omega ,t)$$ . The coefficients $$a,b,c,{\bar{b}}$$ are uniformly bounded, and a satisfies ellipticity condition. The random initial data $$u_0 = u_0(\omega ,x)$$ is nonnegative. To establish the $$L_p$$ -regularity theory, we impose an algebraic condition on $$\lambda $$ depending on the nonlinearity of the diffusion coefficient $$\sigma (u)$$ . For example, if $$\sigma (u)$$ has Lipschitz continuity, linear growth, and boundedness in u, $$\lambda $$ is assumed to be less than or equal to 1; $$\lambda \in (0,1]$$ . However, if $$\sigma (u) = |u|^{1+\lambda _0}$$ with $$\lambda _0\in [0,1/2)$$ , $$\lambda $$ is taken to be less than 1; $$\lambda \in (0,1)$$ . Under those conditions, the uniqueness, existence, and regularity of the solution are obtained in stochastic $$L_p$$ spaces. Also, we have the maximal Hölder regularity by employing the Hölder embedding theorem. For example, if $$\lambda \in (0,1]$$ and $$\sigma (u)$$ has Lipschitz continuity, linear growth, and boundedness in u, for $$T<\infty $$ and $$\varepsilon >0$$ , $$\begin{aligned} u \in C^{1/4 - \varepsilon ,1/2 - \varepsilon }_{t,x}([0,T]\times {\mathbb {R}})\quad (a.s.). \end{aligned}$$ On the other hand, if $$\lambda \in (0,1)$$ and $$\sigma (u) = |u|^{1+\lambda _0}$$ with $$\lambda _0\in [0,1/2)$$ , for $$T<\infty $$ and $$\varepsilon >0$$ , $$\begin{aligned} u \in C^{\frac{1/2-(\lambda -1/2) \vee \lambda _0}{2} - \varepsilon ,1/2-(\lambda -1/2) \vee \lambda _0 - \varepsilon }_{t,x}([0,T]\times {\mathbb {R}})\quad (a.s.). \end{aligned}$$ It should be noted that if $$\sigma (u)$$ is bounded Lipschitz in u, the Hölder regularity of the solution is independent of $$\lambda $$ . However, if $$\sigma (u)$$ is super-linear in u, the Hölder regularities of the solution are affected by nonlinearities, $$\lambda $$ and $$\lambda _0.$$

Boundary value problems for general first-order elliptic differential operators

We study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary. The adapted boundary operator need not be selfadjoint and the boundary condition need not be pseudo-local.We show the equivalence of various characterisations of elliptic boundary conditions and demonstrate how the boundary conditions traditionally considered in the literature fit in our framework. The regularity of the solutions up to the boundary is proven. We show that imposing elliptic boundary conditions yields a Fredholm operator if the manifold is compact. We provide examples which are conveniently treated by our methods.Link to video abstract: https://vimeo.com/523211595

Regularity for a Nonlinear Discontinuous Subelliptic System with Drift on the Heisenberg Group

In this paper, we prove the partial Hölder regularity of weak solutions and the partial Morrey regularity to horizontal gradients of weak solutions to a nonlinear discontinuous subelliptic system with drift on the Heisenberg group by theA-harmonic approximation, where the coefficients in the nonlinear subelliptic system are discontinuous and satisfy the VMO condition forx, ellipticity and growth condition with the growth index1&lt;p&lt;2for the Heisenberg gradient variable, and the nonhomogeneous terms satisfy the controllable growth condition and the natural growth condition, respectively.

$$L^p$$ theory for the square roots and square functions of elliptic operators having a BMO anti-symmetric part

We consider the operator $$L=-\mathrm{div}(A\nabla )$$ , where A is an $$n\times n$$ matrix of real coefficients and satisfies the ellipticity condition, with $$n\ge 2$$ . We assume that the coefficients of the symmetric part of A are in $$L^\infty ({\mathbb {R}}^n)$$ , and those of the anti-symmetric part of A only belong to the space $$BMO({\mathbb {R}}^n)$$ . We create a complete narrative of the $$L^p$$ theory for the square root of L and show that it satisfies the $$L^p$$ estimates $$\left\| {\sqrt{L}f}\right\| _{L^p}\lesssim \left\| {\nabla f}\right\| _{L^p}$$ for $$1<p<\infty $$ , and $$\left\| {\nabla f}\right\| _{L^p}\lesssim \left\| {\sqrt{L}f}\right\| _{L^p}$$ for $$1<p<2+\epsilon $$ for some $$\epsilon >0$$ depending on the ellipticity constant and the BMO semi-norm of the coefficients. Moreover, we prove the $$L^p$$ estimates for some vertical square functions associated to $$e^{-tL}$$ . In another article of the authors, these results are used to establish the solvability of the Dirichlet problem for elliptic equation $$\mathrm{div}(A(x)\nabla u)=0$$ in the upper half-space $$(x,t)\in {\mathbb {R}}_+^{n+1}$$ with the boundary data in $$L^p({\mathbb {R}}^n,dx)$$ for some $$p\in (1,\infty )$$ .

Asymptotic spreading for Fisher-KPP reaction-diffusion equations with heterogeneous shifting diffusivity

&lt;p style='text-indent:20px;'&gt;We determine the asymptotic spreading speed of the solutions of a Fisher-KPP reaction-diffusion equation, starting from compactly supported initial data, when the diffusion coefficient is a fixed bounded monotone profile that is shifted at a given forcing speed and satisfies a general uniform ellipticity condition. Depending on the monotonicity of the profile, we are able to characterize this spreading speed as a function of the forcing speed and the two linear spreading speeds associated to the asymptotic problems at &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ x = \pm \infty $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. Most notably, when the profile of the diffusion coefficient is increasing we show that there is an intermediate range for the forcing speed where spreading actually occurs at a speed which is larger than the linear speed associated with the homogeneous state around the position of the front. We complement our study with the construction of strictly monotone traveling front solutions with strong exponential decay near the unstable state when the profile of the diffusion coefficient is decreasing and in the regime where the forcing speed is precisely the selected spreading speed.&lt;/p&gt;

Extrinsic Black Hole Uniqueness in Pure Lovelock Gravity

We define a notion of extrinsic black hole in pure Lovelock gravity of degree $k$ which captures the essential features of the so-called Lovelock-Schwarzschild solutions, viewed as rotationally invariant hypersurfaces with null $2k$-mean curvature in Euclidean space $\mathbb R^{n+1}$, $2\leq 2k\leq n-1$. We then combine a regularity argument with a rigidity result by Ara\'ujo-Leite to prove, under a natural ellipticity condition, a global uniqueness theorem for this class of black holes. As a consequence we obtain, in the context of the corresponding Penrose inequality for graphs established by Ge-Wang-Wu, a local rigidity result for the Lovelock-Schwarzschild solutions.

On the maximum principles and the quantitative version of the Hopf lemma for uniformly elliptic integro-differential operators

In the present paper we prove estimates on subsolutions of the equation − A v + c ( x ) v = 0 , x ∈ D , where D ⊂ R d is a domain (i.e. an open and connected set) and A is an integro-differential operator of the Waldenfels type, whose differential part satisfies the uniform ellipticity condition on compact sets. In general, the coefficients of the operator need not be continuous but only bounded and Borel measurable. Some of our results may be considered “quantitative” versions of the Hopf lemma, as they provide the lower bound on the outward normal directional derivative at the maximum point of a subsolution in terms of its value at the point. We shall also show lower bounds on the subsolution around its maximum point by the principal eigenfunction associated with A and the domain. Additional results, among them the weak and strong maximum principles, the weak Harnack inequality are also proven.

Strong ellipticity conditions and infinitesimal stability within nonlinear strain gradient elasticity

We discuss connections between the strong ellipticity condition and the infinitesimal instability within the nonlinear strain gradient elasticity. The strong ellipticity (SE) condition describes the property of equations of statics whereas the infinitesimal stability is introduced as the positive definiteness of the second variation of an energy functional. Here we establish few implications which simplify the further analysis of stability using formulated SE conditions. The results could be useful for the analysis of solutions of homogenized models of beam-lattice materials at different scales.

The p-ellipticity condition for second order elliptic systems and applications to the Lamé and homogenization problems

The notion of p-ellipticity has recently played a significant role in improving our understanding of issues of solvability of boundary value problems for scalar complex valued elliptic PDEs. In particular, the presence of p-ellipticity ensures higher regularity of solutions of such equations.In this work we extend the notion of p-ellipticity to second order elliptic systems. Recall that for systems, there is no single notion of ellipticity, rather a more complicated picture emerges with ellipticity conditions of varying strength such as the Legendre, Legendre-Hadamard and integral conditions. A similar picture emerges when p-ellipticity is considered. In this paper, we define three new notions of p-ellipticity, establish relationships between them and show that each of them does play an important role in solving boundary value problems.These important roles are demonstrated by establishing extrapolation results for solvability of the Lp Dirichlet problem for elliptic systems, followed by applications of this result in two different scenarios: one for the Lamé system of linear elasticity and another in the theory of homogenization.

On local Lipschitz regularity for quasilinear equations in the Heisenberg group

The goal of this article is to establish local Lipschitz continuity of solutions for a class of sub-elliptic equations of divergence form, in the Heisenberg Group. The considered hypothesis for the growth and ellipticity condition is a natural generalization of the sub-elliptic p-Laplace equation and more general quasilinear equations with polynomial or exponential type growth.

An exact solution to the Lame problem for a hollow sphere for new types of nonlinear elastic materials in the case of large deformations

Constitutive relations of two classes are proposed for nonlinear elastic isotropic materials, which, in case of purely volumetric deformation, are reduced to the Murnaghan’s equation of state. Exact analytical solution of the Lame problem of the radially symmetric deformation of a hollow sphere is obtained for one of these material classes. Nonlinear effects are studied. The non-uniqueness of solution is obtained for the case in which the sphere radii are specified in the initial configuration. It is shown for this case that there is a limiting pressure, above which the problem has no solution. The strong ellipticity conditions are tested. The obtained results can be used in geomechanics for modeling the recrystallization of metamorphic rocks.

Layer potential theory for the anisotropic Stokes system with variable L∞ symmetrically elliptic tensor coefficient

The aim of this paper is to develop a layer potential theory in L2‐based weighted Sobolev spaces on Lipschitz bounded and exterior domains of , n ≥ 3, for the anisotropic Stokes system with L∞ viscosity tensor coefficient satisfying an ellipticity condition for symmetric matrices with zero matrix trace. To do this, we explore equivalent mixed variational formulations and prove the well‐posedness of some transmission problems for the anisotropic Stokes system in Lipschitz domains of , with the given data in L2‐based weighted Sobolev spaces. These results are used to define the volume (Newtonian) and layer potentials and to obtain their properties. Then, we analyze the well‐posedness of the exterior Dirichlet and Neumann problems for the anisotropic Stokes system with L∞ symmetrically elliptic tensor coefficient by representing their solutions in terms of the obtained volume and layer potentials.

Fu–Yau Hessian equations

We solve the Fu–Yau equation for arbitrary dimension and arbitrary slope $\alpha^\prime$. Actually we obtain at the same time a solution of the open case $\alpha^\prime \gt 0$, an improved solution of the known case $\alpha^\prime \lt 0$, and solutions for a family of Hessian equations which includes the Fu–Yau equation as a special case. The method is based on the introduction of a more stringent ellipticity condition than the usual $\Gamma_k$ admissible cone condition, and which can be shown to be preserved by precise estimates with scale.

Macro- and micro-instabilities in incompressible bioinspired composite materials with nacre-like microstructure

In this work, we study the macroscopic and microscopic instabilities in 2D incompressible nacre-like composite materials induced by uniaxial loadings. The interchange between macro- and micro-instabilities in bioinspired composite materials has been investigated by examining different combinations of platelets volume fraction, aspect ratio, and shear modulus contrast between the stiff (platelets) and soft (matrix) phases. It has been highlighted that the critical instability mode shapes, together with their critical load factors and wavelengths, are highly influenced by the microscopic geometrical arrangement and the material composition. For a wide range of geometrical and material parameters, the instabilities are found to be global with an in-phase mode shape. It has been shown that the out-of-phase instability mode shapes occur at a significantly higher magnitude of the homogenized energy compared to the in-phase ones. The results indicate that, adopting a small unit cell assembly, the microscopic stability analysis provides in most of cases strong underestimates of the critical stretch ratios, and that an accurate and efficient instability prediction can be instead obtained based on the loss of strong ellipticity condition of the homogenized incremental moduli tensor, except for few cases in which local modes occur.

Half-space Gaussian symmetrization: applications to semilinear elliptic problems

Abstract We consider a class of semilinear equations with an absorption nonlinear zero order term of power type, where elliptic condition is given in terms of Gauss measure. In the case of the superlinear equation we introduce a suitable definition of solutions in order to prove the existence and uniqueness of a solution in ℝ N without growth restrictions at infinity. A comparison result in terms of the half-space Gaussian symmetrized problem is also proved. As an application, we give some estimates in measure of the growth of the solution near the boundary of its support for sublinear equations. Finally we generalize our results to problems with a nonlinear zero order term not necessary of power type.

Eigenvalue estimates via weighted L2-technique

Under various elliptic boundary conditions, we obtain lower eigenvalue estimates for Dirac operators by using the weighted L2-technique which has its origin in the Morrey-Kohn-Hörmander theory from several complex variables. Lower bounds in terms of the volume of the underlying manifolds are also deduced from the sharp Sobolev inequality.

Programmable sufficient conditions for the strong ellipticity of partially symmetric tensors

The condition for strong ellipticity of the equilibrium equations plays a significant role in the theory of elasticity. For isotropic elastic materials and anisotropic linearly elastic materials, identification of the strong ellipticity conditions for the corresponding equilibrium equations has been discussed in many references and obtained some equivalent checkable criteria. But for general nonlinearly elastic materials, it is hardly possible to give checkable equivalent criteria for the strong ellipticity condition of the associated equilibrium equations. In 2009, Qi et al. pointed that the strong ellipticity condition of the equilibrium equations can be equivalently transformed into the strong ellipticity condition of partially symmetric tensors. In this paper, using the M-eigenvalues of partially symmetric tensors, we give some easily computable and verifiable sufficient conditions for the strong ellipticity of partially symmetric tensors. Based on these criteria, some direct algorithms for identifying the strong ellipticity condition of partially symmetric tensors are derived. Numerical examples show that the proposed criteria are efficient in identifying the strong ellipticity condition of the equilibrium equations, especially for general nonlinearly elastic materials.

Failure of Fredholm solvability for the Dirichlet problem corresponding to weakly elliptic systems

It is known (cf. Martell et al. in Rev Mat Iberoam 32(3):913–970, 2016) that the $$L^p$$ -Dirichlet problem for constant coefficient second-order systems satisfying the Legendre-Hadamard (strong) ellipticity condition is well posed in the upper half-space. However, this may fail if only weak ellipticity for the system in question is assumed. In this paper we shall show that the aforementioned failure is at a fundamental level, in the sense that there exist systems which are weakly elliptic (i.e., their characteristic matrix is invertible) for which the $$L^p$$ -Dirichlet problem in the upper half-space is not even Fredholm solvable.