Laplace Equation

Show that \(u(x)=\frac {1}{4\pi |x|}\) is the fundamental solution of the Laplace equation in \(\mathbb {R}^3\), \(-\Delta u=\delta _{0}\) in the sense of distributions, i.e. \[ \int _{\mathbb {R}^3} u(x)\Delta \varphi (x)\,dx=-\varphi (0) \]

for all \(\varphi \) smooth and compactly supported. You should provide a fully detailed proof, not just a formal computation.

Suppose someone treats one of the variables in the Laplace equation as time and tries to solve the evolution problem \[ \begin {aligned} u_{tt}+u_{xx}&=0\quad (0<x<l,\ t>0),\\ u(0,t)&=u(l,t)=0,\qquad u(x,0)=\phi (x),\qquad u_t(x,0)=\psi (x), \end {aligned} \] similar to the wave equation. Show that this problem has no continuous dependence on data, even if the time \(t\) belongs to a finite interval \(0<t<T\) (\(T\) is a positive constant).

Formulate and prove the maximum principle for Laplace’s equation \(u_{xx}+u_{yy}=0\).

Suppose that \(u(x)\in C^2(D)\) is a harmonic function on the smooth bounded domain \(D\subset \mathbb {R}^n\) so that \(\nabla ^2u=0\) in \(D\).

1. prove the mean value property: if \(x\in D\) and \(r>0\) is chosen so that the ball of radius \(r\) and center \(x\) satisfies \(B_r(x)\subset D\) then \[ u(x)=\frac {1}{\omega _nr^{n-1}}\int _{\partial B_r(x)}u(y)\dd y \] where \(\omega _n\) is the \(n-1\) dimensional area of \(\partial B_1\). Hence show that \[ u(x)=\frac {n}{\omega _nr^n}\int _{B_r(x)}u(z)\dd z. \]
2. Assuming that \(D\) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in D\) only if \(u\) is constant.

Solve Laplace’s equation \(\phi _{xx}+\phi _{yy}=0\) in the domain \[ D=\{(x,y)\in \mathbb {R}^2:x^2+y^2>1,\ (x+2)^2+y^2<9\} \] with boundary conditions: \(\phi (x,y)=a\) when \(x^2+y^2=1\) and \(\phi (x,y)=b\) when \((x+2)^2+y^2=9\).

Suppose that \(u(x)\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^n\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(x\in \Omega \) and \(r>0\) is chosen such that \(B_r(x)\subset \Omega \) (ball of radius \(r\) centered at \(x\)) then \[ u(x)=\frac {1}{\omega _nr^{n-1}}\int _{\partial B_r(x)}u(s)\dd s, \] where \(\omega _n\) is the measure of \(\partial B_1\). Hence show that \[ u(x)\le \frac {n}{\omega _nr^n}\int _{B_r(x)}u(y)\dd y. \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

Find the real-valued function \(\Phi (z)\) that solves Laplace’s equation in the region between the two noncentric circles \(\lvert z\rvert =1\) and \(\lvert z-1\rvert =\frac {5}{2}\) subject to the boundary conditions \(\Phi =0\) on \(\lvert z\rvert =1\) and \(\Phi =1\) on \(\lvert z-1\rvert =\frac {5}{2}\).

The fundamental solution for the Laplacian in two dimensional space is \(G(\mathbf {x},\mathbf {y})=\frac {1}{2\pi }\ln (\abs {\mathbf {x}-\mathbf {y}})\), where \(\mathbf {x}=(x_1,x_2)\). What is the Green’s function for the Laplacian operator on the first quadrant \(0<x_1,x_2<\infty \), subject to boundary conditions \(G(x_1,0)=0\), \(G_{x_1}(0,x_2)=0\)? Be sure to verify the validity of your solution.

1. Prove: If a function is analytic then its real and imaginary parts are harmonic.
2. Prove: If a function \(u(x,y)\) is harmonic in a domain \(D\), then \(u(x,y)\) cannot attain a strict local maximum in \(D\). [Hint: If \(f(z)=u+iv\), then function \(\phi (z)=e^{f(z)}\) has absolute value \(\lvert \phi (z)\rvert =e^u\).]

1. Consider a 2-dimensional map \((x,y)\to (u,v)\). Explain: If the map is given by an analytic function [\(u+iv=f(x+iy)\)] and at some point \(z_0=x_0+iy_0\) the derivative \(f'(z_0)\ne 0\), then this map preserves small shapes in the vicinity of \(z_0\).
2. Is a square conformally equivalent to a circle?
3. Explain: A map by analytic function preserves Laplace’s equation.

Solve Laplace’s equation \[ u_{xx}+u_{yy}=0 \quad \text {in the domain}\quad D=\{(x,y)\in \mathbb {R}^2:\ (x-1)^2+y^2>1\ \& \ (x-2)^2+y^2<4\} \] subject to the boundary condition \[ u(x,y)=a\text { when }(x-1)^2+y^2=1 \qquad \text {and}\qquad u(x,y)=b\text { when }(x-2)^2+y^2=4 \] [\(a\) and \(b\) are real parameters; the two circles touch each other; express your answer in terms of the original real variables \(x,y\)].

Suppose that \(u(x)\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^n\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(x\in \Omega \) and \(r>0\) is chosen such that \(B_r(x)\subset \Omega \) (ball of radius \(r\) centered at \(x\)) then \[ u(x)=\frac {1}{\omega _n r^{n-1}}\int _{\partial B_r(x)}u(s)\dd s, \] where \(\omega _n\) is the measure of \(\partial B_1\). Hence show that \[ u(x)\le \frac {n}{\omega _n r^n}\int _{B_r(x)}u(y)\dd y. \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

Solve Laplace’s equation \[ \phi _{xx}+\phi _{yy}=0 \quad \text {in the domain}\quad D=\{z:\lvert z-1\rvert >1,\ \lvert z-2\rvert <2\} \] subject to the boundary condition \[ \phi (x,y)=a\text { when }\lvert z-1\rvert =1 \qquad \text {and}\qquad \phi (x,y)=b\text { when }\lvert z-2\rvert =2 \] (\(a\) and \(b\) are real parameters).

Suppose that \(u(x)\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^n\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(x\in \Omega \) and \(r>0\) is chosen such that \(B_r(x)\subset \Omega \) (ball of radius \(r\) centered at \(x\)) then \[ u(x)=\frac {1}{\omega _n r^{n-1}}\int _{\partial B_r(x)}u(s)\dd s, \] where \(\omega _n\) is the measure of \(\partial B_1\). Hence show that \[ u(x)\le \frac {n}{\omega _n r^n}\int _{B_r(x)}u(y)\dd y. \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

Solve Laplace’s equation \[ \phi _{xx}+\phi _{yy}=0 \quad \text {in the domain}\quad D=\{(x,y)\in \mathbb {R}^2:\ x^2+y^2>1,\ x^2+(y+1)^2<4\} \] subject to the boundary condition \[ \phi (x,y)=a\text { when }x^2+y^2=1 \qquad \text {and}\qquad \phi (x,y)=b\text { when }x^2+(y+1)^2=4 \] (\(a\) and \(b\) are real parameters; the two circles touch each other; express your answer in terms of the original real variables \(x\) ans \(y\)).

A function \(u\in C^2(\Omega )\), \(\Omega \subset \mathbb {R}^n\), is subharmonic (superharmonic) in \(\Omega \) if \(\nabla ^2u\ge 0\) (\(\nabla ^2u\le 0\)) in \(\Omega \).

1. Show that if \(u\) is subharmonic then for every ball \(B_R(\mathbf {x})\Subset \mathbf {R}\) \[ u(\mathbf {x})\le \frac {1}{\omega _n R^{n-1}}\int _{\partial B_R(\mathbf {x})}u(\sigma )\dd \sigma . \] Hence show that \[ u(\mathbf {x})\le \frac {n}{\omega _n R^n}\int _{B_R(\mathbf {x})}u(\mathbf {y})\dd y. \] How are the inequalities changed if \(u\) is superharmonic?
2. Show that if \(u\in C(\overline {\Omega })\) is subharmonic (superharmonic), the maximum (minimum) of \(u\) is attained only on \(\partial \Omega \), unless \(u\) is constant.
3. Show that if \(u\) is harmonic in \(\Omega \) then \(u^2\) is subharmonic.
4. Let \(u\) be subharmonic in \(\Omega \) and \(F:\mathbb {R}\to \mathbb {R}\) smooth. Under what conditions on \(F\) is \(F(u)\) subharmonic?

Suppose that \(u(\mathbf {x})\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^n\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(\mathbf {x}\in \Omega \) and \(r>0\) is chosen such that \(B_r(\mathbf {x})\subset \Omega \) (ball of radius \(r\) centered at \(\mathbf {x}\)) then \[ u(\mathbf {x})=\frac {1}{\omega _n r^{n-1}}\int _{\partial B_r(\mathbf {x})}u(s)\dd s, \] where \(\omega _n\) is the measure of \(\partial B_1\). Hence show that \[ u(\mathbf {x})\le \frac {n}{\omega _n r^n}\int _{B_r(\mathbf {x})}u(\mathbf {y})\dd y. \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

Suppose that \(u(\mathbf {x})\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^n\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(\mathbf {x}\in \Omega \) and \(r>0\) is chosen such that \(B_r(\mathbf {x})\subset \Omega \) (ball of radius \(r\) centered at \(\mathbf {x}\)) then \[ u(\mathbf {x})=\frac {1}{\omega _n r^{n-1}}\int _{\partial B_r(\mathbf {x})}u(s)\dd s, \] where \(\omega _n\) is the measure of \(\partial B_1\). Hence show that \[ u(\mathbf {x})\le \frac {n}{\omega _n r^n}\int _{B_r(\mathbf {x})}u(\mathbf {y})\dd y. \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

Suppose that \(u(x_1,x_2)\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^2\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(y=(y_1,y_2)\in \Omega \) and \(r>0\) is chosen such that \(B_r(y)\subset \Omega \) (ball of radius \(r\) centered at \(y\)) then \[ u(y_1,y_2)=\frac {1}{2\pi }\int _0^{2\pi }u(y_1+r\cos \theta ,y_2+r\sin \theta )\dd \theta . \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

1. For a smooth domain \(D\subset \mathbb {R}^n\), solve Poisson’s equation \[ \nabla ^2u=f\text { in }D,\qquad \frac {\partial u}{\partial n}=g\text { on }\partial D \] in terms of an appropriately defined Green’s function.
2. Show that the two-dimensional Green’s function for the Laplacian in an unbounded domain is \[ G(x,x')=\frac {1}{2\pi }\ln \lvert x-x'\rvert . \]
3. Use the method of images to derive the Green’s function for the Laplacian in the half-space \(x\in \mathbb {R}\), \(y>0\) with a Neumann boundary condition on \(y=0\), and evaluate the corresponding solution of part (a) when \(f=0\).

Suppose that \(u(x_1,x_2)\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^2\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(y=(y_1,y_2)\in \Omega \) and \(r>0\) is chosen such that \(B_r(y)\subset \Omega \) (ball of radius \(r\) centered at \(y\)) then \[ u(y_1,y_2)=\frac {1}{2\pi }\int _0^{2\pi }u(y_1+r\cos \theta ,y_2+r\sin \theta )\dd \theta . \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

Find a solution \(u(x,y)\) of Laplace’s equation on the domain \(-\infty <x<\infty \), \(0<y<\infty \) for which \(u(x,0)=x^{1/2}\) for \(0<x<\infty \). What is \(u(x,0)\) for \(-\infty <x<0\)?

Suppose that \(u(x_1,x_2)\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^2\), so \(\Delta u=0\) in \(\Omega \).

1. Prove the mean value property: if \(y=(y_1,y_2)\in \Omega \) and \(r>0\) is chosen such that \(B_r(y)\) (the ball of radius \(r\), centered at \(y\)) satisfies \(B_r(y)\subset \Omega \), then \[ u(y_1,y_2)=\frac {1}{2\pi }\int _0^{2\pi } u(y_1+r\cos \theta ,y_2+r\sin \theta )\dd \theta . \]
2. Assuming \(\Omega \) is connected, prove that \(u\) can attain a maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.

1. Verify that the real and imaginery parts of a complex analytic function satisfy Laplace’s equation.
2. Give an interpretation of the complex function \[ w(z)=z+\frac {a^2}{z} \] as a flow past some object. That is, determine the streamlines for this function. Where are the stagnation points?

1. Find the spherically symmetric eigenfunctions and corresponding eigenvalues for the Laplacian on a spherical domain of radius \(R\) with Dirichlet boundary conditions \(u(R)=0\).
2. Use these eigenfunctions to solve the heat equation on a spherical domain of radius \(R\), with \(u(R,t)=U_0\) and \(u(r,0)=0\).
3. Estimate the time \(t\) at which \(u(0,t)\) is \(\frac {U_0}{2}\). How does this time depend on the radius \(R\)?

1. Solve Poisson’s equation \[ \nabla ^2u=f\text { in }D,\qquad u=g\text { on }\partial D \] in terms of an appropriately defined Green’s function.
2. Show that the two–dimensional Green’s function for the Laplacian in an unbounded domain is \[ G(\mathbf {x},\mathbf {x}')=\frac {1}{2\pi }\ln \lvert \mathbf {x}-\mathbf {x}'\rvert . \]
3. Use the method of images to derive the Green’s function for the Laplacian in the half–space \(x\in \mathbb {R}\), \(y>0\) with a Dirichlet boundary condition on \(y=0\), and evaluate the corresponding solution of part (a) when \(f=0\).

Find a solution \(u(x,y)\) of Laplace’s equation in 2 dimensions for which \(u(x,0)=|x|^{1/2}\).

Solve Laplace’s equation \(\Delta \phi =0\) in the domain between the two nonconcentric circles \(|z|=1\) and \(|z-1|=5/2\), subject to the boundary conditions \(\phi =a\) on \(|z|=1\) and \(\phi =b\) on \(|z-1|=5/2\).

Solve the Laplace equation \(\Delta \phi =0\) subject to the boundary conditions: \[ \begin {aligned} \phi& =a &&\text {on } \{z: |z|=2,\ z\ne 2\},\\ \phi& =b &&\text {on } \{z: |z-1|=1,\ z\ne 2\}. \end {aligned} \] (\(a\) and \(b\) are real constants; \(z=x+iy\).)

Let \(K:H\to H\) be a compact linear operator on the infinite-dimensional Hilbert space \(H\).

1. Prove that \(K^{-1}\), if it exists, is not bounded.
2. Given \(T>0\), let \(H=L^2(0,T)\). For \(u\in H\), define \[ (Ku)(t)=\int _0^T e^{-st}u(s)\,ds,\qquad 0\le t\le T. \] Prove that \(K:H\to H\) is a compact operator. What are the consequences of this fact for the utility of the Laplace transform as a numerical method?

Consider the wave equation \[ \frac {\partial ^2\phi }{\partial t^2}=\frac {\partial ^2\phi }{\partial x^2}, \qquad 0\le x\le l \] with the following boundary conditions \[ \begin {gathered} \phi (x,0)=0,\qquad \frac {\partial \phi }{\partial t}(x,0)=0,\\ \phi (0,t)=0,\qquad \phi (l,t)=f(t). \end {gathered} \]

1. Show that the Laplace transform of the solution \(\widehat {\Phi }(x,s)\) is given by \[ \widehat {\Phi }(x,s)=\frac {\widehat {F}(s)\sinh sx}{\sinh sl} \] where \(\widehat {F}(s)\) is the Laplace transform of \(f(t)\).
2. Suppose that \(\phi _0(x,t)\) is the solution when \(f(t)=1\). Show that the solution for general \(f\) is given by \[ \phi (x,t)=\int _0^t \frac {\partial \phi _0}{\partial t'}(x,t')f(t-t')\,dt'. \]

Laplace Equation. Let \(\Omega \subset \mathbb {R}^n\) be open. Suppose \(u\in C(\Omega )\) satisfies the mean value property: for every ball \(B_r(\xi )=\{x\in \mathbb {R}^n:\lvert x-\xi \rvert <r\}\) with \(B_r(\xi )\subset \Omega \), \[ u(\xi )=\frac {1}{\omega _n}\int _{\lvert x\rvert =1}u(\xi +rx)\,dS_x, \] where \(\omega _n\) is the measure of the \((n-1)\)-dimensional unit sphere in \(\mathbb {R}^n\). Prove that \(\Delta u=0\) in \(\Omega \).

Laplace Equation. Let \(\Omega \subset \mathbb {R}^n\) be a domain. Let \(\Omega _1\) be a bounded subdomain with \(\overline {\Omega }_1\subset \Omega \).

1. State the Harnack Inequality for harmonic functions.
2. Let \(\{u_n\}\) be a monotone increasing sequence of nonnegative harmonic functions on \(\Omega \). Suppose that for some point \(\xi \in \Omega _1\), the sequence of real numbers \(\{u_n(\xi )\}\) is bounded. Prove that \(\{u_n\}\) converges uniformly on \(\Omega _1\) to a harmonic function. You may use standard theorems about convergence of sequences of harmonic functions without proof.

Obtain the inverse Laplace transform of the function \[ \widehat {F}(s)=s^{-a},\qquad 0<a<1 \] by evaluating the associated Bromwich contour integral with a branch cut along the negative real axis. Express the answer in terms of the gamma function \[ \Gamma (z)=\int _0^\infty t^{z-1}e^{-t}\,dt. \]

Find the solution of Laplace’s equation on the circular domain \(0\le r\le R\), \(0\le \theta <2\pi \), with Dirichlet boundary data \(u(R,\theta )=f(\theta )\).

Consider the wave equation \[ \frac {\partial ^2\phi }{\partial t^2}=\frac {\partial ^2\phi }{\partial x^2},\qquad 0\le x\le l \] with the following boundary conditions \[ \begin {gathered} \phi (x,0)=0,\qquad \frac {\partial \phi }{\partial t}(x,0)=0,\\ \phi (0,t)=0,\qquad \phi (l,t)=1. \end {gathered} \]

1. Show that the Laplace transform of the solution \(\widehat {\Phi }(x,s)\) is given by \[ \widehat {\Phi }(x,s)=\frac {\sinh sx}{s\sinh sl}. \]
2. Obtain the solution \(\phi (x,t)\) by inverting the Laplace transform using a Bromich integral and show that it can be expressed as the infinite series \[ \phi (x,t)=\frac {x}{l}+\sum _{n=1}^{\infty }\frac {2(-1)^n}{n\pi } \sin \left (\frac {n\pi x}{l}\right )\cos \left (\frac {n\pi t}{l}\right ). \]

Find the bounded solution \(u(x,y)\) of the Laplace equation in the sector \[ D=\{(r,\theta ): 0<\theta <3\pi /4\}\qquad (x+iy=z=re^{i\theta }) \] satisfying the boundary conditions \[ u=0\text { when }\theta =0,\qquad u=5\text { when }\theta =\frac {3\pi }{4}. \]

Show that there is a constant \(c<\infty \) depending only on \(n\) and \(r\) so that \[ \sup \{\lvert Du(x)\rvert :x\in \overline {B_r(0)}\}\le c\sup \{\lvert u(x)\rvert :x\in \partial B_{2r}(0)\}. \] for all harmonic functions \(u\) on \(B_{2r}(0)\).

Show that \(w_\varphi \) is harmonic in \(B_r(\xi )\).

Suppose \(f(\theta )\) is a \(2\pi \) periodic function. Define its Hilbert Transform as follows: Let \(u(r,\theta )\) be the solution of Laplace’s equation on the unit circle satisfying Neumann data \(u_r(1,\theta )=f(\theta )\). The Hilbert transform of \(f\) is \(H(f)=u_\theta (1,\theta )\).

Use separation of variables to find the Hilbert Transform of \(f\) in terms of its Fourier coefficients.

Prove by means of the Poisson Formula that harmonic functions are of class \(C^\infty \).

Evaluate \(\Delta _3(1/r^2)\), \(r^2=x^2+y^2+z^2\) in \(\mathbb {R}^3\), where \(\Delta _3\) is the distributional Laplacian.