no sensible extension of the concept of traces to for since any bounded linear operator which extends the classical trace must be zero on the space of test functions , which is a dense subset of , implying that such an operator would be zero everywhere. Generalized normal trace Let denote the distributional divergence of a vector field . For and bounded Lipschitz domain define which is a Banach space with norm . Let denote the exterior unit normal field on . Then there exists a bounded linear operator , where is the conjugate exponent to and denotes the continuous dual space to a Banach space , such that extends the normal trace for in the sense that . The value of the normal trace operator for is defined by application of the divergence theorem to the vector field where is the trace extension operator from above. Application. Any weak solution to in a bounded Lipschitz domain has a normal derivative in the sense of . This follows as since and . This result is notable since in Lipschitz domains in general , such that may not lie in the domain of the trace operator . Application The theorems presented above allow a closer investigation of the boundary value problem on a Lipschitz domain from the motivation. Since only the Hilbert space case is investigated here, the notation is used to denote etc. As stated in the motivation, a weak solution to this equation must satisfy and for all , where the right-hand side must be interpreted for as a duality product with the value . Existence and uniqueness of weak solutions The characterization of the range of implies that for to hold the regularity is necessary. This regularity is also sufficient for the existence of a weak solution, which can be seen as follows. By the trace extension theorem there exists such that . Defining by we have that and thus by the characterization of as space of trace zero. The function then satisfies the integral equation for all . Thus the problem with inhomogeneous boundary values for could be reduced to a problem with homogeneous boundary values for , a technique which can be applied to any linear differential equation. By the Riesz representation theorem there exists a unique solution to this problem. By uniqueness of the decomposition , this is equivalent to the existence of a unique