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Given an elliptic differential operator L of second order with smoothcoefficients in a bounded domain with smooth boundary. We show that if thecoefficients are H\"older-continuous up to the boundary and the boundary is$C^{2,\alpha}$-smooth that on the space of all $C^{2,\alpha}$-smooth (up to theboundary) functions u fulfilling both u=0 and Lu=0 (on the boundary) theoperator L is dissipative and closable to an generator of a strong continuousoperator semigroup in the space of continuous functions with zero boundarycondition. Moreover we show that if the coefficients of the second order andfirst order derivatives are in $C^{2,\alpha}$ then the above mentioned subspaceof $C^{2,\alpha}$ which is a core for the generator of the semigroup, can beembedded (continued) in a contractive and smooth way. Thus we construct alinear extension operator which maps a $C^{2,\alpha}$-smooth functionfulfilling the boundary conditions to a $C^{2,\alpha}$-smooth function in$\R^n$ such that the supremum is not increased.

Core property of smooth contractive embeddable functions for an elliptic operator