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On bounded domains, the unshifted equation is an identity-minus-compact equation

Statement

Assume Countable Choice. Let $Ω \subseteq R^{n}$ be open, let $μ \geq β$ and let $K_{μ}$ be the shifted solution operator of The shifted elliptic solution operator for the form $a$ of Uniformly elliptic divergence-form operators and their sesquilinear forms. For $f \in L^{2} (Ω)$ and $u \in H_{0}^{1} (Ω)$ the following are equivalent:

$a (u, v) = (f, v)_{L^{2}}$ for every $v \in H_{0}^{1} (Ω)$ , i.e. $u$ is a weak Dirichlet solution in the sense of Weak Dirichlet solutions for a divergence-form operator;
$(I - μ K_{μ}) u = K_{μ} f$ as elements of $L^{2} (Ω)$ , where $I$ is the identity of $L^{2} (Ω)$ .

Both sides of (2) lie in $H_{0}^{1} (Ω)$ . Thus for arbitrary open $Ω$ the weak equation is equivalent to this identity-minus-bounded-operator equation. If $Ω$ is bounded, also assume the Axiom of Choice; then $K_{μ}$ on $L^{2} (Ω)$ , and hence $μ K_{μ}$ , is compact by The shifted solution operator is compact on $L^{2}$ , so (2) is an identity-minus-compact Fredholm equation. The algebraic equivalence applies to the general divergence-form operator, including nonsymmetric lower-order terms.

Facts & Assumptions

Given: Countable Choice; an open set $Ω \subseteq R^{n}$ ; a fixed $μ \geq β$ ; the shifted solution operator $K_{μ}$ of the general divergence-form operator; $f \in L^{2} (Ω)$ and $u \in H_{0}^{1} (Ω)$ .

[F1]

Definition of $K_{μ}$ : for $g \in L^{2} (Ω)$ , $K_{μ} g$ is the unique class in $H_{0}^{1} (Ω)$ with $a_{μ} (K_{μ} g, v) = (g, v)_{L^{2}}$ for all $v \in H_{0}^{1} (Ω)$ , where $a_{μ} = a + μ (\cdot, \cdot)_{L^{2}}$ ; $K_{μ}$ is linear and maps $L^{2} (Ω)$ into $H_{0}^{1} (Ω)$ (The shifted elliptic solution operator, Zero-boundary Sobolev space as a norm closure).

[F2]

Weak Dirichlet solutions: $u$ is a weak solution of $a (u, v) = (f, v)_{L^{2}}$ for every $v \in H_{0}^{1} (Ω)$ exactly when the identity holds for all test classes $v \in H_{0}^{1} (Ω)$ with datum $f \in L^{2} (Ω)$ (Weak Dirichlet solutions for a divergence-form operator, Uniformly elliptic divergence-form operators and their sesquilinear forms).

[F3]

Compactness: if $Ω$ is bounded and the Axiom of Choice holds, the $L^{2}$ realization of $K_{μ}$ is compact, hence so is $μ K_{μ}$ , and $I - μ K_{μ}$ is an identity-minus-compact operator on $L^{2} (Ω)$ (The shifted solution operator is compact on $L^{2}$ , The space $L^{p} (μ)$ as the quotient by null functions, The Axiom of Choice).

Proof

technique · direct

1.1F1F2givenalgebra

Equivalence. Condition (1) says $a (u, v) = (f, v)_{L^{2}}$ for all $v \in H_{0}^{1} (Ω)$ . Adding $μ (u, v)_{L^{2}}$ to both sides, this is equivalent to $a_{μ} (u, v) = (f + μ u, v)_{L^{2}}$ for all $v \in H_{0}^{1} (Ω)$ , where $f + μ u \in L^{2} (Ω)$ . By the defining uniqueness clause of [F1] for the datum $f + μ u$ , this holds exactly when $u = K_{μ} (f + μ u)$ in $H_{0}^{1} (Ω)$ . Rearranging the linear identity gives $u - μ K_{μ} u = K_{μ} f$ , that is $(I - μ K_{μ}) u = K_{μ} f$ in $L^{2} (Ω)$ , and conversely the same rearrangement recovers the defining identity for $f + μ u$ and hence condition (1). No symmetry of $a$ and no sign condition on the lower-order coefficients is used.

1.2F1givenalgebra

Location of the two sides. Since $u \in H_{0}^{1} (Ω)$ by hypothesis and $K_{μ}$ maps $L^{2} (Ω)$ into $H_{0}^{1} (Ω)$ , both $u$ and $μ K_{μ} u$ , hence both sides of (2), lie in $H_{0}^{1} (Ω)$ ( $μ K_{μ} u \in H_{0}^{1}$ because $H_{0}^{1}$ is a linear subspace); the equality itself is an equality of $L^{2}$ classes.

2.1F3step 1.1step 1.2given∎

Compact case. If $Ω$ is bounded and the Axiom of Choice is assumed, [F3] makes the $L^{2}$ realization of $μ K_{μ}$ compact, so (2) is the equation $(I - C) u = K_{μ} f$ with $C : = μ K_{μ}$ compact, an identity-minus-compact equation; for unbounded $Ω$ the equivalence of step 1.1 remains valid as an identity-minus-bounded-operator equation and no compactness or Fredholm claim is made.

Depends on

Used by

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Sources

John K. Hunter, Notes on Partial Differential Equations (UC Davis, revised 18 June 2014, complete 242-page notes) (standard reference, not scraped)
Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (Springer Universitext, 2011, complete 614-page text) (standard reference, not scraped)