# Tagged Questions

Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

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### How Coulomb Gauge guarantees uniqueness in regard to Lax-Milgram Lemma, curl-curl problem

The Lax-Milgram lemma gives insight on existence and uniqueness of a PDE of the type $$a(u,v)=f(v)$$ Positive definiteness and coercivity are required for the bilinear form $a(u,v)$. In the curl-...
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### Are harmonic functions ($\Delta u=0$) with compact support unique?

Does anyone how to solve the problem $$\Delta u=0$$ with $u \in R^n$ and $u(x) \to 0$ as $|x| \to \infty$? Is $u=0$ the only solution? Many thanks.
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### Dealing with the logarithm of absolute value when solving $u_t+xu_x=1$ by method of characteristics

I was trying to come up with a solution to the PDE IVP problem $u_t + x u_x =1$, $u(x,0) = f(x)$. Here's how I went about it: We can the following relations by applying the Method of Characteristics: ...
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### Solving large set of PDEs and verfication of solution

Please comment on solution of following system of PDEs, I have checked solution several times but could not find any error but there are problems when I proceed with solution obtained as below. I ...
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### PDE realated to heat equation with exponential additive term

I want to solve a PDE realated to heat equation with exponential additive term $${\partial u\over\partial t}={1\over x^2}{\partial\over\partial x}(x^2{\partial u\over\partial x})+e^u$$ I dervived ...
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### Existence and Uniqueness of Minimization problem in Sobolev space

"Consider the functional $$F(u) = \int_{\Omega}[ \frac{1}{2} \nabla u(x) |^2 + g(x) u(x)] dx$$ and the set K = \{ v \in H^1 (\Omega): v = 0\ \text{on} \ \partial \Omega\ in\ the \ sense\ ...
Given the problem$-\Delta u(x,y)=f(x,y)$ on unit rectangle $\Omega=[0,1]^{2}$ and $u(x,y)=g(x,y)$ on $\partial\Omega$, what is the finite difference matrix associated with step size $h=1/(2N+1)$ where ...
Can someone please help me to correct the following inequality: Let $a,b,c$ be three strictely positif numbers we have : \$a^{\frac{1}{108}}b^{\frac{3}{4}}c^6\le \frac{1}{27}a^{\frac{1}{4}}+\frac{...