Question:
take example:such $f(x,y)$ in the unit disc $ D$ that extends continuously to ∂D
such that $f_{xx}'',f''_{yy}$ exist,and continuously,But $f''_{xy}(0,0)$ is not exsit.
My try: I have consider sometimes,and I can't looking for this function $f(x,y)$ such this condition.
can anyone can help me? Thank you
"Extends continuously to $\partial D$" is quite irrelevant: we can always restrict the domain to a neighborhood of $(0,0)$ and rescale.
Here is one example. Let $$f(x,y)= xy\sqrt{-\log(x^2+y^2)},\quad f(0,0)=0$$ Since the restriction of $f$ to either coordinate axis is identically zero, we have $f_{xx}(0,0)=0$ and $f_{yy}(0,0)=0$. Also, the partial derivative $$f_{xx}(x,y)=\frac{xy}{x^2+y^2} \cdot \frac{x^2-(x^2+3y^2)\log(x^2+y^2)}{(x^2+y^2) \log(x^2+y^2)}\cdot \frac{1}{\sqrt{-\log(x^2+y^2)}}$$ tends to $0$ as $(x,y)\to (0,0)$, because the first two fractions are bounded. Thus, $f_{xx}$ is continuous. By symmetry, so is $f_{yy}$.
Yet, $f_x(x,0)=y\sqrt{-\log(y^2)}$ is not differentiable with respect to $y$ at $y=0$. Thus, $f_{xy}(0,0)$ does not exist. Neither does $f_{yx}(0,0)$, again by symmetry.
This kind of counterexample comes up in elliptic PDEs. See
