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Chris Judge
Chris Judge
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Let A_t$A_t$ be family of second order, positive, elliptic differential operator mapping Sobolev H^2$H^2$ of a compact smooth manifold (or bounded domain) to L^2. Suppose that the coefficients of A_t$A_t$ converge uniformly in C^k$C^k$ for every k$k$ to the coefficients of a second order, positive, elliptic differential operator A$A$. A$A$ is invertible (with domain L^2 and range H^2) and so we may consider the sequence A_t \circ A_0^{-1}$A_t \circ A_0^{-1}$ of operators from $L^2$ to $L^2$. Does this family converge to the identity in the L^2$L^2$ operator norm? Why or why not?

Let A_t be family of second order, positive, elliptic differential operator mapping Sobolev H^2 of a compact smooth manifold (or bounded domain) to L^2. Suppose that the coefficients of A_t converge uniformly in C^k for every k to the coefficients of a second order, positive, elliptic differential operator A. A is invertible (with domain L^2 and range H^2) and so we may consider the sequence A_t \circ A_0^{-1}. Does this family converge to the identity in the L^2 operator norm? Why or why not?

Let $A_t$ be family of second order, positive, elliptic differential operator mapping Sobolev $H^2$ of a compact smooth manifold (or bounded domain) to L^2. Suppose that the coefficients of $A_t$ converge uniformly in $C^k$ for every $k$ to the coefficients of a second order, positive, elliptic differential operator $A$. $A$ is invertible (with domain L^2 and range H^2) and so we may consider the sequence $A_t \circ A_0^{-1}$ of operators from $L^2$ to $L^2$. Does this family converge to the identity in the $L^2$ operator norm? Why or why not?

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Chris Judge
Chris Judge
  • 494
  • 4
  • 13

Convergence of elliptic operators

Let A_t be family of second order, positive, elliptic differential operator mapping Sobolev H^2 of a compact smooth manifold (or bounded domain) to L^2. Suppose that the coefficients of A_t converge uniformly in C^k for every k to the coefficients of a second order, positive, elliptic differential operator A. A is invertible (with domain L^2 and range H^2) and so we may consider the sequence A_t \circ A_0^{-1}. Does this family converge to the identity in the L^2 operator norm? Why or why not?