It's well known that:Given a continuous function defined on the boundary, there exists a unique harmonic function defined on the disk. What if we replace the disk by a cone?

Take $\Omega$ as $$z^2=x^2+y^2,x^2+y^2\leq1$$ for example.

What's the regularity of boundary function f that can guarantee the existence of a harmonic function of u? If u exists, what's the regularity of u? Does u satisfy the Green's formula? ie $$\int_\Omega |\nabla u|^2dxdy=\int_{x^2+y^2=1}u\frac{\partial u}{\partial n}dS?$$

It's well known that: Given a continuous function defined on the boundary of the disk, then there exists a unique harmonic function in the interior of the disk. What if we replace the disk by a cone?

Take $\Omega$ as $$z^2=x^2+y^2,x^2+y^2\leq1$$ for example.

What's the regularity of boundary function $f$ that can guarantee the existence of a harmonic function of $u$? If $u$ exists, what's the regularity of $u$? Does $u$ satisfy the Green's formula? i.e., $$\int_\Omega |\nabla u|^2dxdy=\int_{x^2+y^2=1}u\frac{\partial u}{\partial n}dS?$$

It's well known that:Given a continuous function defined on the boundary, there exists a unique harmonic function defined on the disk. What if we replace the disk by a cone?

Take $\Omega$ as $$z=x^2+y^2,x^2+y^2\leq1$$ for example.

What's the regularity of boundary function f that can guarantee the existence of a harmonic function of u? If u exists, what's the regularity of u? Does u satisfy the Green's formula? ie $$\int_\Omega |\nabla u|^2dxdy=\int_{x^2+y^2=1}u\frac{\partial u}{\partial n}dS?$$

It's well known that:Given a continuous function defined on the boundary, there exists a unique harmonic function defined on the disk. What if we replace the disk by a cone?

Take $\Omega$ as $$z^2=x^2+y^2,x^2+y^2\leq1$$ for example.

What's the regularity of boundary function f that can guarantee the existence of a harmonic function of u? If u exists, what's the regularity of u? Does u satisfy the Green's formula? ie $$\int_\Omega |\nabla u|^2dxdy=\int_{x^2+y^2=1}u\frac{\partial u}{\partial n}dS?$$

It's well known that:Given a continuous function defined on the boundary, there exists a unique harmonic function defined on the disk. What if we replace the disk by a cone?

Take $\Omega$ as $$z=x^2+y^2,x^2+y^2\leq1$$ for example.

What's the regularity of boundary function f that can guarantee the existence of a harmonic function of u? If u exists, what's the regularity of u? $$|\nabla u|^2=(\frac{\partial u}{\partial r})^2+\frac {1}{r^2}(\frac{\partial u}{\partial \theta})^2?$$ Does u satisfy the Green's formula? ie $$\int_\Omega |\nabla u|^2dxdy=\int_{x^2+y^2=1}u\frac{\partial u}{\partial n}dS?$$

It's well known that:Given a continuous function defined on the boundary, there exists a unique harmonic function defined on the disk. What if we replace the disk by a cone?

Take $\Omega$ as $$z=x^2+y^2,x^2+y^2\leq1$$ for example.

What's the regularity of boundary function f that can guarantee the existence of a harmonic function of u? If u exists, what's the regularity of u? Does u satisfy the Green's formula? ie $$\int_\Omega |\nabla u|^2dxdy=\int_{x^2+y^2=1}u\frac{\partial u}{\partial n}dS?$$