## On the injectivity and linearization of the coefficient-to-solution mapping for elliptic boundary value problems.(English)Zbl 0817.35021

The authors study the coefficient-to-solution mapping $$a\to a(u)$$, where $$u$$ is a solution of $$\text{div} (a \text{ grad } u)=f$$ in $$\Omega\in \mathbb{R}^ n$$, $$n\leq 3$$ with $$\Omega$$ a bounded domain. A priori estimates for the coefficient $$a$$ in terms of $$u$$ and injectivity of the mapping $$a\to a(u)$$ are derived. The linearization of the coefficient-to-solution mapping is also investigated.

### MSC:

 35J25 Boundary value problems for second-order elliptic equations 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs

### Keywords:

coefficient-to-solution mapping; injectivity; linearization
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