Suppose that f : ( 0, ∞ ) × ( 0, ∞ ) → ( 0, ∞ ) is a continuous symmetric function.
(1) where (T>0), (A t)) is a continuous symmetric matrix of order N and (F: [0,T]times{mathbb{R}}^{N}rightarrow mathbb{R}) is locally Lipschitz continuous in x and (partial F t,x)) denotes the Clarke subdifferential of F for x.
Seek (w in V) that satisfies begin{aligned} a w,upsilon)=b f,upsilon), quad forallupsilonin V. end{aligned} (2.2) It is obvious that (a(cdot,cdot)) is a continuous, symmetric, and V-elliptic bilinear function on (Vtimes V) and (b f,cdot)) is a linear and continuous function on V (see [13]).
It is obvious that (a(cdot,cdot)) is a continuous, symmetric, and V-elliptic bilinear form on (Vtimes V) and (b f,cdot)) is a continuous and linear functional on V. Thus, we can use (Vert cdot Vert _{a}=sqrt{a(cdot,cdot)}) as a norm in V which is equivalent to the norm (Vert cdot Vert _{2}) induced by (H^{2}(Omega)) and we know from the Lax-Milgram theorem that (6) has a unique solution.
Let be a symmetric harmonic convex set with nonempty interior, and let be a continuous symmetry function on.
