Year: 2024
Type: article
Abstract: In mathematical physics, the gradient operator with nonconstant coefficients encompasses various models, including Fourier’s law for heat propagation and Fick’s first law, that relates the diffusive flux to the gradient of the concentration. Specifically, consider $$n\ge 3$$ n ≥ 3 orthogonal unit vectors $$e_1,\ldots ,e_n\in {\mathbb {R}}^n$$ e 1 , … , e n ∈ R n , and let $$\Omega \subseteq {\mathbb {R}}^n$$ Ω ⊆ R n be some (in general unbounded) Lipschitz domain. This paper investigates the spectral properties of the gradient operator $$T=\sum _{i=1}^ne_ia_i(x)\frac{\partial }{\partial x_i}$$ T = ∑ i = 1 n e i a i ( x ) ∂ ∂ x i with nonconstant positive coefficients $$a_i:{\overline{\Omega }}\rightarrow (0,\infty )$$ a i : Ω ¯ → ( 0 , ∞ ) . Under certain regularity and growth conditions on the $$a_i$$ a i , we identify bisectorial or strip-type regions that belong to the S -resolvent set of T . Moreover, we obtain suitable estimates of the associated resolvent operator. Our focus lies in the spectral theory on the S -spectrum, designed to study the operators acting in Clifford modules V over the Clifford algebra $${\mathbb {R}}_n$$ R n , with vector operators being a specific crucial subclass. The spectral properties related to the S -spectrum of T are linked to the inversion of the operator $$Q_s(T):=T^2-2s_0T+|s|^2$$ Q s ( T ) : = T 2 - 2 s 0 T +
Authors: F. Colombo, F. Mantovani, P. Schlosser
Institution: Politecnico di Milano
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4402679229