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We consider second-order partial differential operators $H$ in divergence form on $Ri^d$ with a positive-semidefinite, symmetric, matrix $C$ of real $L_infty$-coefficients and establish that $H$ is strongly elliptic if and only if the associated semigroup kernel satisfies local lower bounds, or, if and only if the kernel satisfies Gaussian upper and lower bounds.
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In this paper, we establish two sufficient conditions for the strong ellipticity of any fourth-order elasticity tensor and investigate a class of tensors satisfying the strong ellipticity condition, the elasticity $mathscr{M}$-tensor. The first sufficient condition is that the strong ellipticity holds if the unfolding matrix of this fourth-order elasticity tensor can be modified into a positive definite one by preserving the summations of some corresponding entries. Second, an alternating projection algorithm is proposed to verify whether an elasticity tensor satisfies the first condition or not. Besides, the elasticity $mathscr{M}$-tensor is defined with respect to the M-eigenvalues of elasticity tensors. We prove that any nonsingular elasticity $mathscr{M}$-tensor satisfies the strong ellipticity condition by employing a Perron-Frobenius-type theorem for M-spectral radii of nonnegative elasticity tensors. Other equivalent definitions of nonsingular elasticity $mathscr{M}$-tensors are also established.
We study high-order harmonic generation (HHG) from aligned molecules in strong elliptically polarized laser fields numerically and analytically. Our simulations show that the spectra and polarization of HHG depend strongly on the molecular alignment and the laser ellipticity. In particular, for small laser ellipticity, large ellipticity of harmonics with high intensity is observed for parallel alignment, with forming a striking ellipticity hump around the threshold. We show that the interplay of the molecular structure and two-dimensional electron motion plays an important role here. This phenomenon can be used to generate bright elliptically-polarized EUV pulses.
We provide sufficient conditions on the coefficients of a stochastic evolution equation on a Hilbert space of functions driven by a cylindrical Wiener process ensuring that its mild solution is positive if the initial datum is positive. As an application, we discuss the positivity of forward rates in the Heath-Jarrow-Morton model via Musielas stochastic PDE.
For solutions of ${rm div},(DF(Du))=f$ we show that the quasiconformality of $zmapsto DF(z)$ is the key property leading to the Sobolev regularity of the stress field $DF(Du)$, in relation with the summability of $f$. This class of nonlinearities encodes in a general way the notion of uniform ellipticity and encompasses all known instances where the stress field is known to be Sobolev regular. We provide examples showing the optimality of this assumption and present three applications: the study of the strong locality of the operator ${rm div},(DF(Du))$, a nonlinear Cordes condition for equations in divergence form, and some partial results on the $C^{p}$-conjecture.
We prove a maximum principle for mild solutions to stochastic evolution equations with (locally) Lipschitz coefficients and Wiener noise on weighted $L^2$ spaces. As an application, we provide sufficient conditions for the positivity of forward rates in the Heath-Jarrow-Morton model, considering the associated Musiela SPDE on a homogeneous weighted Sobolev space.
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