[Paper Review] Criterion for the $L^{p}$-dissipativity of second order differential operators with complex coefficients

Alberto Cialdea, Vladimir Maz’ya|ArXiv.org|Dec 11, 2004

Spectral Theory in Mathematical Physics4 citations

TL;DR

This paper establishes a necessary and sufficient algebraic condition for the $L^p$-dissipativity of second-order elliptic differential operators with complex coefficients. The key result is that the condition $|p-2|\,|\langle\mathop{\mathscr{I}m}\nolimits\mathop{\mathscr{A}}\nolimits\xi,\xi\rangle| \leq 2\sqrt{p-1}\,\langle\mathop{\mathscr{R}e}\nolimits\mathop{\mathscr{A}}\nolimits\xi,\xi\rangle$ for all $\xi \in \mathbb{R}^n$ characterizes $L^p$-dissipativity when the imaginary part of the coefficient matrix $\mathop{\mathscr{A}}\nolimits$ is symmetric, and $L^p$-quasi-dissipativity (and quasi-contractivity of the associated semigroup) when it is not, even with lower-order terms present.

ABSTRACT

We prove that the algebraic condition $|p-2| |< {\mathscr Im}{\mathscr A}ξ,ξ>| \leq 2 \sqrt{p-1} < {\mathscr Re}{\mathscr A}ξ,ξ>$ (for any $ξ\in\mathbb{R}^{n}$) is necessary and sufficient for the $L^{p}$-dissipativity of the Dirichlet problem for the differential operator $ abla^{t}({\mathscr A} abla)$, where ${\mathscr A}$ is a matrix whose entries are complex measures and whose imaginary part is symmetric. This result is new even for smooth coefficients, when it implies a criterion for the $L^{p}$-contractivity of the corresponding semigroup. We consider also the operator $ abla^{t}({\mathscr A} abla)+{\bf b} abla +a$, where the coefficients are smooth and ${\mathscr Im}{\mathscr A}$ may be not symmetric. We show that the previous algebraic condition is necessary and sufficient for the $L^{p}$-quasi-dissipativity of this operator. The same condition is necessary and sufficient for the $L^{p}$-quasi-contractivity of the corresponding semigroup. We give a necessary and sufficient condition for the $L^{p}$-dissipativity in $\mathbb{R}^{n}$ of the operator $ abla^{t}({\mathscr A} abla)+{\bf b} abla +a$ with constant coefficients.

Motivation & Objective

• To establish a necessary and sufficient algebraic condition for $L^p$-dissipativity of the Dirichlet problem for second-order differential operators with complex coefficients.
• To extend the criterion to operators with lower-order terms, especially when the imaginary part of the coefficient matrix is not symmetric.
• To characterize $L^p$-quasi-dissipativity and quasi-contractivity of the associated semigroups in the general case of smooth coefficients.
• To provide a complete characterization for constant coefficient operators in $\mathbb{R}^n$.

Proposed method

• Derive the $L^p$-dissipativity condition via analysis of the associated quadratic form $\mathop{\mathscr{L}}\nolimits(u,v) = \int_\Omega \langle\mathop{\mathscr{A}}\nolimits\nabla u, \nabla v\rangle\,dx$.
• Use spectral and variational techniques to relate the dissipativity of the operator to the coercivity of the real part and the boundedness of the imaginary part of the coefficient matrix.
• Apply the theory of $C_0$-semigroups and duality to link $L^p$-dissipativity of an operator and its adjoint to the generation of contraction or quasi-contraction semigroups.
• Introduce the concept of $L^p$-quasi-dissipativity to handle cases where the imaginary part of $\mathop{\mathscr{A}}\nolimits$ is not symmetric, by shifting the operator by a real constant.
• Reduce the general case with lower-order terms to an equivalent problem involving the symmetric part of $\mathop{\mathscr{A}}\nolimits$ to apply the main criterion.
• Establish the equivalence between $L^p$-dissipativity of the form and $L^p$-dissipativity of the operator via density and closedness arguments in $L^p$ spaces.

Experimental results

Research questions

• RQ1 What algebraic condition on the coefficient matrix $\mathop{\mathscr{A}}\nolimits$ ensures $L^p$-dissipativity of the Dirichlet problem for a second-order elliptic operator with complex coefficients?
• RQ2 How does the condition change when the imaginary part of $\mathop{\mathscr{A}}\nolimits$ is not symmetric, and what alternative notion of dissipativity applies?
• RQ3 Can the criterion be extended to operators with lower-order terms, and under what conditions is the associated semigroup $L^p$-quasi-contractive?
• RQ4 What is the precise range of $p$ for which the semigroup generated by such an operator is $L^p$-quasi-contractive?
• RQ5 Is there a characterization of $L^p$-dissipativity for constant coefficient operators on $\mathbb{R}^n$?

Key findings

• The condition $|p-2|\,|\langle\mathop{\mathscr{I}m}\nolimits\mathop{\mathscr{A}}\nolimits\xi,\xi\rangle| \leq 2\sqrt{p-1}\,\langle\mathop{\mathscr{R}e}\nolimits\mathop{\mathscr{A}}\nolimits\xi,\xi\rangle$ for all $\xi \in \mathbb{R}^n$ is necessary and sufficient for $L^p$-dissipativity of the Dirichlet problem when $\mathop{\mathscr{I}m}\nolimits\mathop{\mathscr{A}}\nolimits$ is symmetric.
• When $\mathop{\mathscr{I}m}\nolimits\mathop{\mathscr{A}}\nolimits$ is not symmetric, the same algebraic condition is necessary and sufficient for $L^p$-quasi-dissipativity of the operator $\nabla^t(\mathop{\mathscr{A}}\nolimits\nabla) + \mathbf{b}\nabla + a$.
• The same condition ensures $L^p$-quasi-contractivity of the semigroup generated by the operator, meaning the semigroup generated by $A - \omega I$ is contractive for some $\omega > 0$.
• For constant coefficient operators on $\mathbb{R}^n$, the condition provides a necessary and sufficient criterion for $L^p$-dissipativity.
• The range of $p$ for which the semigroup is $L^p$-quasi-contractive is determined by the infimum of the ratio $\frac{\langle\mathop{\mathscr{R}e}\nolimits\mathop{\mathscr{A}}\nolimits(x)\xi,\xi\rangle}{|\langle\mathop{\mathscr{I}m}\nolimits\mathop{\mathscr{A}}\nolimits(x)\xi,\xi\rangle|}$, leading to explicit bounds on $p$.
• The result is new even for smooth coefficients, and provides a complete algebraic characterization of $L^p$-contractivity for the associated semigroup.