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Severin Schraven
Severin Schraven
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Let $u\in C^2(\Omega)$ with $\Omega = B_1(0)\subseteq \mathbb{R}^d$. Assume that $u$ is subharmonic and satisfies the inequality $$ \Delta u(x) \geq C>0 $$ for all $x\in \Omega$. Furthermore, we know that $u\geq 0$. Is there a reasonable lower bound for $\sup_\Omega u$ in terms of $C$?

By the strong maximum principle for subharmonic functions we get that the supremum of $u$ on $\Omega$ cannot be attained in the interior (the function cannot be constant as this would violate the inequality $-\Delta u >0$$\Delta u >0$). In fact, we cannot even have a local maximum by the same argument. So morally I would expect that $u$ has some directions in which is grows quite fast and I was wondering whether there is a quantitative way of describing this (respectively whether it is true to begin with).

Let $u\in C^2(\Omega)$ with $\Omega = B_1(0)\subseteq \mathbb{R}^d$. Assume that $u$ is subharmonic and satisfies the inequality $$ \Delta u(x) \geq C>0 $$ for all $x\in \Omega$. Furthermore, we know that $u\geq 0$. Is there a reasonable lower bound for $\sup_\Omega u$ in terms of $C$?

By the strong maximum principle for subharmonic functions we get that the supremum of $u$ on $\Omega$ cannot be attained in the interior (the function cannot be constant as this would violate the inequality $-\Delta u >0$). In fact, we cannot even have a local maximum by the same argument. So morally I would expect that $u$ has some directions in which is grows quite fast and I was wondering whether there is a quantitative way of describing this (respectively whether it is true to begin with).

Let $u\in C^2(\Omega)$ with $\Omega = B_1(0)\subseteq \mathbb{R}^d$. Assume that $u$ is subharmonic and satisfies the inequality $$ \Delta u(x) \geq C>0 $$ for all $x\in \Omega$. Furthermore, we know that $u\geq 0$. Is there a reasonable lower bound for $\sup_\Omega u$ in terms of $C$?

By the strong maximum principle for subharmonic functions we get that the supremum of $u$ on $\Omega$ cannot be attained in the interior (the function cannot be constant as this would violate the inequality $\Delta u >0$). In fact, we cannot even have a local maximum by the same argument. So morally I would expect that $u$ has some directions in which is grows quite fast and I was wondering whether there is a quantitative way of describing this (respectively whether it is true to begin with).

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Severin Schraven
Severin Schraven
  • 1k
  • 7
  • 17

Growth of nonnegative functions satisfying $-\Delta$\Delta u \geq C>0$

Let $u\in C^2(\Omega)$ with $\Omega = B_1(0)\subseteq \mathbb{R}^d$. Assume that $u$ is subharmonic and satisfies the inequality $$ -\Delta u(x) \geq C>0 $$$$ \Delta u(x) \geq C>0 $$ for all $x\in \Omega$. Furthermore, we know that $u\geq 0$. Is there a reasonable lower bound for $\sup_\Omega u$ in terms of $C$?

By the strong maximum principle for subharmonic functions we get that the supremum of $u$ on $\Omega$ cannot be attained in the interior (the function cannot be constant as this would violate the inequality $-\Delta u >0$). In fact, we cannot even have a local maximum by the same argument. So morally I would expect that $u$ has some directions in which is grows quite fast and I was wondering whether there is a quantitative way of describing this (respectively whether it is true to begin with).

Growth of nonnegative functions satisfying $-\Delta u \geq C>0$

Let $u\in C^2(\Omega)$ with $\Omega = B_1(0)\subseteq \mathbb{R}^d$. Assume that $u$ is subharmonic and satisfies the inequality $$ -\Delta u(x) \geq C>0 $$ for all $x\in \Omega$. Furthermore, we know that $u\geq 0$. Is there a reasonable lower bound for $\sup_\Omega u$ in terms of $C$?

By the strong maximum principle for subharmonic functions we get that the supremum of $u$ on $\Omega$ cannot be attained in the interior (the function cannot be constant as this would violate the inequality $-\Delta u >0$). In fact, we cannot even have a local maximum by the same argument. So morally I would expect that $u$ has some directions in which is grows quite fast and I was wondering whether there is a quantitative way of describing this (respectively whether it is true to begin with).

Growth of nonnegative functions satisfying $\Delta u \geq C>0$

Let $u\in C^2(\Omega)$ with $\Omega = B_1(0)\subseteq \mathbb{R}^d$. Assume that $u$ is subharmonic and satisfies the inequality $$ \Delta u(x) \geq C>0 $$ for all $x\in \Omega$. Furthermore, we know that $u\geq 0$. Is there a reasonable lower bound for $\sup_\Omega u$ in terms of $C$?

By the strong maximum principle for subharmonic functions we get that the supremum of $u$ on $\Omega$ cannot be attained in the interior (the function cannot be constant as this would violate the inequality $-\Delta u >0$). In fact, we cannot even have a local maximum by the same argument. So morally I would expect that $u$ has some directions in which is grows quite fast and I was wondering whether there is a quantitative way of describing this (respectively whether it is true to begin with).

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Severin Schraven
Severin Schraven
  • 1k
  • 7
  • 17

Growth of nonnegative functions satisfying $-\Delta u \geq C>0$

Let $u\in C^2(\Omega)$ with $\Omega = B_1(0)\subseteq \mathbb{R}^d$. Assume that $u$ is subharmonic and satisfies the inequality $$ -\Delta u(x) \geq C>0 $$ for all $x\in \Omega$. Furthermore, we know that $u\geq 0$. Is there a reasonable lower bound for $\sup_\Omega u$ in terms of $C$?

By the strong maximum principle for subharmonic functions we get that the supremum of $u$ on $\Omega$ cannot be attained in the interior (the function cannot be constant as this would violate the inequality $-\Delta u >0$). In fact, we cannot even have a local maximum by the same argument. So morally I would expect that $u$ has some directions in which is grows quite fast and I was wondering whether there is a quantitative way of describing this (respectively whether it is true to begin with).