Strictly monotone operator ($u\ne v⟹⟨A(u)-A(v),u-v⟩>0$)

A strictly monotone operator $A:X\to X^{\ast }$ on a real Banach space satisfies strict operator monotonicity for every pair of distinct inputs. It is injective: $Au=Av$ would make the pairing zero and hence force $u=v$. In particular $Au=f$ has at most one solution for each $f\in X^{\ast }$.

Let $\Omega$ be a nonempty bounded Lipschitz domain, and define $A:H^1(\Omega )\to H^1(\Omega {)}^{\ast }$ byThis operator is strictly monotone: the difference pairing is the sum of $\Vert \nabla (v-w){\Vert }_2^2$ and the nonnegative integral $(\tanh v-\tanh w)(v-w)$, which is strictly positive wherever $v\ne w$. The bounded-domain hypothesis also makes nonzero constant functions admissible in $H^1$.

For constant $v=t$,Thus $A$ is not a nonlinear coercive operator in the large-norm sense. On $H_0^1(\Omega )$, the Poincaré inequality instead giveswhich proves coercivity on that zero-boundary space. The example separates being a strictly monotone operator from coercivity and identifies the uncontrolled constant direction.

For real vectors $\xi ,\eta$, put $m=(\xi +\eta )/2$ and $d=\xi -\eta$. Expansion givesThis proves strict operator monotonicity of the gradient of $\xi \mapsto |\xi {|}^4/4$. Integrating the inequality controls the difference of two weak gradients in the fourth power of their Lp norm, which yields uniqueness for suitable nonlinear Dirichlet boundary conditions.